In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in the case of a tie, the order is arbitrary). It is noted that each participant won half of their points against the last ten ranked players. How many people participated in the tournament?

Answer: 25


Alice and Bob play a game together as a team on a  $100 \times 100$  board with all unit squares initially white. Alice sets up the game by coloring exactly  $k$  of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least  $10$  red squares and color all of the remaining squares in it red. What is the
smallest  $k$  such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by *Nikola Velov, Macedonia*

Answer: 100


We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?

Answer: \frac{\sqrt{2}}{2}


Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points).

Answer: 376


In the diagram, \(\triangle ABC\) and \(\triangle CDE\) are equilateral triangles. Given that \(\angle EBD = 62^\circ\) and \(\angle AEB = x^\circ\), what is the value of \(x\)?

Answer: 122


In the first quarter, an automobile factory fulfilled $25\%$ of the annual plan for car production. The number of cars produced in the second, third, and fourth quarters was proportional to the numbers $11, 25, 12,$ and $13,5$. Determine the percentage by which the annual plan was exceeded, given that in the second quarter, the factory produced $1.08$ times more products than in the first quarter.

Answer: 13.2


A cylinder with a radius of $20 \text{ cm}$ touches the ground along the line $e$. Perpendicular to the line $e$, a $50 \text{ cm}$ long rod is placed resting against the cylinder, such that the end of the rod on the ground is $40 \text{ cm}$ away from the line $e$. How high is the other end of the rod?

Answer: 40\,


In the convex quadrilateral \(ABCD\), \(AC \perp BD\), \(\angle BCA = 10^\circ\), \(\angle BDA = 20^\circ\), \(\angle BAC = 40^\circ\). Find \(\angle BDC\).

Answer: 60


If the product of two numbers is an odd number, why is the sum of these numbers an even number?

Answer: 2


Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\frac{2S_n+8}{a_n+3}$ for all positive integers $n$.

Answer: \frac{5}{2}


John walks from home to school with a constant speed, and his sister Joan bikes twice as fast. The distance between their home and school is 3 km. If Joan leaves home 15 minutes after John, then they arrive at school at the same time. What is the walking speed (in km/h) of John?

Answer: 6\,


Given the equation \(x^{2}+2xy+y^{2}+3x+y=0\) represents a parabola in the \(xOy\) plane, find the equation of its axis of symmetry.

Answer: x+y+1=0


In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$.

Answer: \frac{\sqrt{3}}{3}


Find the smallest natural number \( k \) such that for some natural number \( a \), greater than 500,000, and some natural number \( b \), the equation \(\frac{1}{a} + \frac{1}{a+k} = \frac{1}{b}\) holds.

Answer: 1001


In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) respectively in the ratio \(\frac{AM}{AC} = \frac{CM}{CE} = r\). If the points \( B \), \( M \), and \( N \) are collinear, find the value of \( r \).

Answer: \frac{\sqrt{3}}{3}


Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk than the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red.

Suppose the bottom level has 2048 disks of which 2014 are red. What is the colour of the disk at the top?

Answer: blue


Let  $V$  be the set of all continuous functions  $f\colon [0,1]\to \mathbb{R}$ , differentiable on  $(0,1)$ , with the property that  $f(0)=0$  and  $f(1)=1$ . Determine all  $\alpha \in \mathbb{R}$  such that for every  $f\in V$ , there exists some  $\xi \in (0,1)$  such that
\[f(\xi)+\alpha = f'(\xi)\]

Answer: \frac{1}{1}


Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

Answer: f(x) = x


Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
$\textbf{(A)} ~\frac{12}{13} \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~\frac{13}{12} \qquad\textbf{(D)}   ~\frac{24}{13} \qquad\textbf{(E)} ~2$

Answer: \textbf{(A)}~\frac{12}{13}


We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$.

What is the length of the broken line?

Answer: \sqrt{2}


4. Suppose that  $\overline{A2021B}$  is a six-digit integer divisible by  $9$ . Find the maximum possible value of  $A \cdot B$ .

5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?

6. Suppose that  $a$  and  $b$  are positive integers such that  $\frac{a}{b-20}$  and  $\frac{b+21}{a}$  are positive integers. Find the maximum possible value of  $a + b$ .

Answer: 143


Determine the sum of coefficients $A$, $B$, $C$, and $D$ for the simplified polynomial form of the function
\[ y = \frac{x^3 - 4x^2 - 9x + 36}{x - 3} \]
which is defined everywhere except at $x = D$.

Answer: -9


Given a triangle $\triangle ABC$, where $2 \sqrt {2}(\sin ^{2}A-\sin ^{2}C)=(a-b)\sin B$, and the radius of the circumcircle is $\sqrt {2}$.
(1) Find $\angle C$;
(2) Find the maximum area of $\triangle ABC$.

Answer: \frac {3 \sqrt {3}}{2}


In January 1859, an eight-year-old boy dropped a newly-hatched eel into a well in Sweden. The eel, named Ale, finally died in August 2014. How many years old was Åle when it died?

Answer: 155


Let $\triangle PQR$ be a triangle in the plane, and let $S$ be a point outside the plane of $\triangle PQR$, so that $SPQR$ is a pyramid whose faces are all triangles.

Suppose that every edge of $SPQR$ has length $18$ or $41$, but no face of $SPQR$ is equilateral. Then what is the surface area of $SPQR$?

Answer: 1440


A conical glass is in the form of a right circular cone. The slant height is  $21$  and the radius of the top rim of the glass is  $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If  $d$  the shortest distance it should crawl to reach the honey drop, what is the integer part of  $d$  ?

[center]![Image](https://i.imgur.com/T1Y3zwR.png)[/center]

Answer: 18


Two milk processing plants must process an equal amount of a certain volume of milk together. The second plant started working $a$ working days later but processed daily $m$ liters more than the first one. After $5a/9$ working days since the start of their joint work, $1/3$ of the entire task remained incomplete. How many working days were required to complete the task if the work was finished simultaneously and each plant processed half of the specified volume of milk?

Answer: 2a


Among  $100$  points in the plane, no three collinear, exactly  $4026$  pairs are connected by line segments. Each point is then randomly assigned an integer from  $1$  to  $100$  inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least  $50$ .

*Proposed by Evan Chen*

Answer: 1037


Let the system of equations be satisfied for positive numbers \( x, y, z \):

\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 108 \\
y^{2} + y z + z^{2} = 49 \\
z^{2} + x z + x^{2} = 157
\end{array}
\right.
\]

Find the value of the expression \( x y + y z + x z \).

Answer: 84


Point \(A\) lies at \((0,4)\) and point \(B\) lies at \((3,8)\). Find the \(x\)-coordinate of the point \(X\) on the \(x\)-axis that maximizes \(\angle AXB\).

Answer: 5\sqrt{2}-3


A computer software retailer has 1200 copies of a new software package to sell. Given that half of them will sell right away at the original price, two-thirds of the remainder will sell later when the price is reduced by 40%, and the remaining copies will sell in a clearance sale at 75% off the original price, determine the original price needed to achieve a total sales revenue of $72000.

Answer: 80.90


Suppose that a rectangle with sides  $ a$  and  $ b$  is arbitrarily cut into  $ n$  squares with sides  $ x_{1},\ldots,x_{n}$ . Show that  $ \frac{x_{i}}{a}\in\mathbb{Q}$   and  $ \frac{x_{i}}{b}\in\mathbb{Q}$  for all  $ i\in\{1,\cdots, n\}$ .

Answer: \frac{x_i}{}\in\mathbb{Q}


Petya plans to spend all 90 days of his vacation in the village, swimming in the lake every second day (i.e., every other day), going to the store for groceries every third day, and solving math problems every fifth day. (On the first day, Petya did all three tasks and got very tired.) How many "pleasant" days will Petya have when he needs to swim but does not need to go to the store or solve problems? How many "boring" days will he have when there are no tasks at all?

Answer: 24


Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.

Answer: 63


Let \( f \) be a function on \([0,1]\) satisfying:
1. \( f(0) = 0, f(1) = 1 \).
2. For all \( x, y \in [0, 1] \), with \( x \leq y \), \( f\left(\frac{x+y}{2}\right) = (1-a) f(x) + a f(y) \), where \( a \) is a real number and \( 0 \leq a \leq 1 \). Find \( f\left(\frac{1}{7}\right) \).

Answer: \frac{1}{7}


Answer: -1


Let $A, B, C, D, E, F$ be 6 points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX=3, BX=2, CY=4, DY=10, EZ=16$, and $FZ=12$, find the perimeter of triangle $XYZ$.

Answer: \frac{77}{6}


A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);[/asy]

Answer: \frac{\sqrt{2} - 1}{2}


In a competition with five participants A, B, C, D, and E, determine the probability that neither B nor C appears adjacent to A.

Answer: \frac{3}{10}


Squares \( A B K L, B C M N, C A O P \) are drawn externally on the sides of a triangle \( A B C \). The line segments \( K L, M N, O P \), when extended, form a triangle \( A^{\prime} B^{\prime} C^{\prime} \). Find the area of \( A^{\prime} B^{\prime} C^{\prime} \) if \( A B C \) is an equilateral triangle of side length 2.

Answer: 12+13\sqrt{3}


Ike and Mike go into a sandwich shop with a total of $$30.00$ to spend. Sandwiches cost $$4.50$ each and soft drinks cost $$1.00$ each. Ike and Mike plan to buy as many sandwiches as they can,
and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how
many items will they buy?
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Answer: \textbf{(D)}9


Let $\alpha$ be an acute angle, and $\cos\left(\alpha+ \frac{\pi}{6}\right) = \frac{3 \sqrt{10}}{10}$, $\tan(\alpha+\beta) = \frac{2}{5}$.
(1) Find the value of $\sin\left(2\alpha+ \frac{\pi}{6}\right)$.
(2) Find the value of $\tan\left(2\beta- \frac{\pi}{3}\right)$.

Answer: \frac{17}{144}


In a convex quadrilateral, the point of intersection of the diagonals divides each of them in a ratio of \(1:2\). The quadrilateral, formed by the consecutive connection of the midpoints of the sides, is a square with a side length of 3. Find the perimeter of the original quadrilateral.

Answer: 4\sqrt{5}+6\sqrt{2}


In an isosceles trapezoid with an acute angle \(\alpha\) at the base, a circle constructed on one of the legs as its diameter is tangent to the other leg.

In what ratio does it divide the larger base of the trapezoid?

Answer: \sin2\alpha:1


Draw a $9 \times 9$ grid on a plane. Fill each small square in the grid with either +1 or -1. A "move" is defined as follows: For any small square, compute the product of the numbers in all the squares that share a common edge with this square (excluding the square itself). After doing this for each square, replace the number in each square with the computed product. The question is whether it is always possible, through a finite number of moves, to make all the numbers in the small squares equal to 1?

Answer: No


In space, two different points \( A \) and \( B \) are used to draw two skew lines \( m \) and \( n \) perpendicular to the line \( AB \). On the line \( m \), a point \( C \) different from \( A \) is selected, and on the line \( n \), a point \( D \) different from \( B \) is selected. Let the length of \( AB \) be \( a \), the length of \( CD \) be \( b \), and the angle between \( m \) and \( n \) be \( \theta \). Determine the radius of the sphere passing through the four points \( A \), \( B \), \( C \), and \( D \).

Answer: \frac{\sqrt{b^2-^2\cos^2\theta}}{2\sin\theta}


Given that \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\) are 7 distinct prime numbers, and the sum of any two of \(a, b, c\) is 800. Let \(d\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \(d\).

Answer: 1594


A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them.

Answer: 174


Given $α \in \left( \frac{π}{2}, π \right)$, and $\sin α = \frac{1}{3}$.

$(1)$ Find the value of $\sin 2α$;
$(2)$ If $\sin (α+β) = -\frac{3}{5}$, and $β \in (0, \frac{π}{2})$, find the value of $\sin β$.

Answer: \frac{6\sqrt{2}+4}{15}


Answer: \frac{17}{224}


Point P lies on the curve represented by the equation $$\sqrt {(x-5)^{2}+y^{2}}- \sqrt {(x+5)^{2}+y^{2}}=6$$. If the y-coordinate of point P is 4, then its x-coordinate is ______.

Answer: x = -3\sqrt{2}


Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$?

Answer: 2 - \sqrt{2}


Answer: -\frac{5}{2}


Arrange the positive integers into two lines as follows:
\begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots
\mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\end{align*} We start with writing  $1$  in the upper line,  $2$  in the lower line and  $3$  again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block.
Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $\ldots$  be the numbers in the upper line. Give an explicit formula for  $a_n$ .

Answer: a_n=F_{n+3}-2


Replace the ellipsis with such a natural number \( p \) so that the answer to the following question is unambiguous: How many lines are drawn on the plane if it is known that they intersect at \( p \) different points?

Answer: 2


Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
 i. Either each of the three cards has a different shape or all three of the cards have the same shape.
 ii. Either each of the three cards has a different color or all three of the cards have the same color.
 iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
 How many different complementary three-card sets are there?

Answer: 117


For what integers  $ n\ge 3$  is it possible to accommodate, in some order, the numbers  $ 1,2,\cdots, n$  in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?

Answer: 3


A circle is inscribed in a square of side $m$, then a square is inscribed in that  circle, then a circle is inscribed in the latter square, and so on.  If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches:
$\textbf{(A)}\ \frac{\pi m^2}{2}\qquad \textbf{(B)}\ \frac{3\pi m^2}{8}\qquad \textbf{(C)}\ \frac{\pi m^2}{3}\qquad \textbf{(D)}\ \frac{\pi m^2}{4}\qquad \textbf{(E)}\ \frac{\pi m^2}{8}$

Answer: \textbf{(A)}\\frac{\pi^2}{2}


Given that \( n \geq 5 \) is a positive integer, and \( a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geq 0 \) with the conditions

\[
\sum_{i=1}^{n} a_{i}^{2} = 1 \quad \text{and} \quad \sum_{i=1}^{n} b_{i} = 1
\]

find the maximum value of

\[
\sum_{i=1}^{n} a_{i}^{1+b_{i}}
\]

Answer: \sqrt{n-1}


Let the set \( M = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). Define the set \( A = \{(x, y, z) \mid x, y, z \in M \text{ and } 9 \mid (x^3 + y^3 + z^3)\} \). Determine the number of elements in the set \( A \).

Answer: 243


Given the sequence \( \{a_n\} \) such that the first term \( a_1 = \frac{1}{2} \) and \( a_{n+1} = 3a_n + 7 \). Find the general term formula for the sequence \( \{a_n\} \).

Answer: a_n=4\cdot3^{n-1}-\frac{7}{2}


Let \( A \) and \( B \) be two moving points on the ellipse \( x^{2}+3 y^{2}=1 \), and \( OA \) is perpendicular to \( OB \) (where \( O \) is the origin). Then, the product of the maximum and minimum values of \( |AB| \) is ______.

Answer: \frac{2 \sqrt{3}}{3}


Some of the 20 metal cubes, identical in size and appearance, are aluminum, while the rest are duralumin (heavier). How can you determine the number of duralumin cubes using 11 weighings on a balance scale with two pans, without using any weights?

Answer: 11


A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remaining chocolates and filled five columns completely, except for one space. He again ate the chocolates from the incomplete column. In the end, one-third of the original number of chocolates remained in the box. Determine:

a) How many chocolates were there in the entire box originally?

b) How many chocolates did Míša eat before the first rearrangement?

Answer: 25


The point values of playing cards in a deck range from 1 to a maximum of 13. Xiao Ming has 3 playing cards with different point values. The sum of the point values of the first and second cards is 25, and the sum of the point values of the second and third cards is 13. What is the point value of the third card?

Answer: 1


Among the points corresponding to number  $1,2,...,2n$  on the real line,  $n$  are colored in blue and  $n$  in red. Let  $a_1,a_2,...,a_n$  be the blue points and  $b_1,b_2,...,b_n$  be the red points. Prove that the sum  $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$  does not depend on coloring , and compute its value. :roll:

Answer: n^2


There is a cube placed on stands and six different paints. How many ways can you paint all the faces of the cube (one color for each face, not necessarily using all the paints) so that adjacent faces (sharing an edge) are painted in different colors? (16 points)

Answer: 4080


What is the ratio of the area of the shaded triangle to the area of the square? The square is divided into a 5x5 grid of smaller, equal-sized squares. A triangle is shaded such that it covers half of a square at the center of the grid and three full squares adjacent to this half-covered square. The vertices of the triangle touch the midpoints of the sides of the squares it covers.

Answer: \frac{7}{50}


On a line, the points \( 0, \pm 1, \pm 2, \pm 3, \ldots \) are marked.

A particle, upon reaching point \( n \), moves to point \( n+1 \) with probability \( \frac{1}{2} \) and to point \( n-1 \) with probability \( \frac{1}{2} \) in one unit of time. Initially, the particle is at point 0. Find:

a) the probability \( x \) that the particle will ever be at point 1;

b) the probability \( y \) that the particle will ever be at point -1;

c) the probability \( z \) that the particle will ever return to point 0 (i.e., be at point 0 at a moment distinct from the initial one).

Answer: 1


How many graphs are there on 10 vertices labeled 1, 2, ..., 10 such that there are exactly 23 edges and no triangles?

Answer: 42840


Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \in N^{*}$, find the value of $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.

Answer: 2015


A hollow silver sphere with an outer diameter of $2 R = 1 \mathrm{dm}$ is exactly half-submerged in water. What is the thickness of the sphere's wall if the specific gravity of silver is $s = 10.5$?

Answer: 0.008


An odd six-digit number is called "just cool" if it consists of digits that are prime numbers, and no two identical digits are adjacent. How many "just cool" numbers exist?

Answer: 729


Points  $A, B,$  and  $C$  are on a circle such that  $\triangle ABC$  is an acute triangle.  $X, Y ,$  and  $Z$  are on the circle such that  $AX$  is perpendicular to  $BC$  at  $D$ ,  $BY$  is perpendicular to  $AC$  at  $E$ , and  $CZ$  is perpendicular to  $AB$  at  $F$ . Find the value of \[ \frac{AX}{AD}+\frac{BY}{BE}+\frac{CZ}{CF},  \] and prove that this value is the same for all possible  $A, B, C$  on the circle such that  $\triangle ABC$  is acute.

[asy]
pathpen = linewidth(0.7);
pair B = (0,0), C = (10,0), A = (2.5,8); path cir = circumcircle(A,B,C);
pair D = foot(A,B,C), E = foot(B,A,C), F = foot(C,A,B), X = IP(D--2*D-A,cir), Y = IP(E--2*E-B,cir), Z = IP(F--2*F-C,cir); D(MP("A",A,N)--MP("B",B,SW)--MP("C",C,SE)--cycle); D(cir); D(A--MP("X",X)); D(B--MP("Y",Y,NE)); D(C--MP("Z",Z,NW)); D(rightanglemark(B,F,C,12)); D(rightanglemark(A,D,B,12)); D(rightanglemark(B,E,C,12));[/asy]

Answer: 4


Given that \( n \) is a positive integer, the real number \( x \) satisfies
$$
|1 - | 2 - |3 - \cdots |(n-1) - |n - x||\cdots||| = x.
$$

Determine the value of \( x \).

Answer: \frac{1}{2}


Given a \(9 \times 9\) chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how many unit squares will contain an even number of marks?

Answer: 56


Answer: 2


If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10} b$, what is the median of the set $\{0, 1, a, b,
1/b\}$?

Answer: b


The perimeter of triangle \( ABC \) is \( 2p \). Points \( M \) and \( N \) are taken on sides \( AB \) and \( AC \) respectively such that \( MN \parallel BC \) and \( MN \) touches the incircle of triangle \( ABC \). Find the maximum length of segment \( MN \).

Answer: \frac{p}{4}


Given $f(x) = x^2$ and $g(x) = |x - 1|$, let $f_1(x) = g(f(x))$, $f_{n+1}(x) = g(f_n(x))$, calculate the number of solutions to the equation $f_{2015}(x) = 1$.

Answer: 2017


Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt{2}}{2}$

Answer: (D)\frac{\sqrt{2}}{2}


A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates?

Answer: 11


Answer: 12+4\sqrt{3}


The chords \(A A'\), \(B B'\), and \(C C'\) of a sphere intersect at a common point \(S\). Find the sum \(S A' + S B' + S C'\), given \(A S = 6\), \(B S = 3\), \(C S = 2\), and the volumes of the pyramids \(SA B C\) and \(S A' B' C'\) are in the ratio \(2: 9\). If the answer is not an integer, round it to the nearest hundredth.

Answer: 18


Suppose \(a\) and \(b\) are positive integers. Isabella and Vidur both fill up an \(a \times b\) table. Isabella fills it up with numbers \(1, 2, \ldots, ab\), putting the numbers \(1, 2, \ldots, b\) in the first row, \(b+1, b+2, \ldots, 2b\) in the second row, and so on. Vidur fills it up like a multiplication table, putting \(ij\) in the cell in row \(i\) and column \(j\). Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is 1200. Compute \(a+b\).

Answer: 21


In the 2011 Zhejiang Province Pilot Module Examination, there were a total of 18 questions. Each examinee was required to choose 6 questions to answer. Examinee A would definitely not choose questions 1, 2, 9, 15, 16, 17, and 18, while Examinee B would definitely not choose questions 3, 9, 15, 16, 17, and 18. Moreover, the 6 questions chosen by A and B had none in common. The total number of ways to select questions that meet these conditions is ______.

Answer: 462


How does the area of the quadrilateral formed by connecting the midpoints of the sides of a concave quadrilateral compare to the area of the original quadrilateral?

Answer: \frac{1}{2}


Two circles \(\omega_{1}\) and \(\omega_{2}\) with radii 10 and 13, respectively, are externally tangent at point \(P\). Another circle \(\omega_{3}\) with radius \(2 \sqrt{2}\) passes through \(P\) and is orthogonal to both \(\omega_{1}\) and \(\omega_{2}\). A fourth circle \(\omega_{4}\), orthogonal to \(\omega_{3}\), is externally tangent to \(\omega_{1}\) and \(\omega_{2}\). Compute the radius of \(\omega_{4}\).

Answer: \frac{92}{61}


Prove that the difference between the sum of the dihedral angles at the edges of a polyhedron and the sum of the solid angles at its vertices is equal to $2 \pi (\Gamma - 2)$, where $\Gamma$ is the number of faces of the polyhedron.

Answer: 2\pi(\Gamma-2)


74. \(ABCD\) is a regular tetrahedron with edge length \(a\). Let \(M\) be the center of face \(ADC\), and \(N\) be the midpoint of edge \(BC\). Find the radius of the sphere inscribed in the trihedral angle at vertex \(A\) and tangent to the line \(MN\).

Answer: \frac{5\sqrt{6}-3\sqrt{2}}{48}


For a sequence \(a_1, a_2, a_3, \ldots\) of real numbers, it is known that
\[ a_n = a_{n-1} + a_{n+2} \quad \text{for} \ n = 2, 3, 4, \ldots \]
What is the largest number of its consecutive elements that can all be positive?

Answer: 5


The solution set to the inequality \(\left|\frac{1}{\log _{\frac{1}{2} x} x}+2\right|>\frac{3}{2}\) is ______.

Answer: (0,1)\cup(1,2^{\frac{2}{7}})\cup(4,+\infty)


In the rectangular coordinate plane, find the number of integer points that satisfy the system of inequalities
\[
\left\{
\begin{array}{l}
y \leqslant 3x \\
y \geqslant \frac{x}{3} \\
x + y \leqslant 100
\end{array}
\right.
\]

Answer: 2551


A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board.

Answer: 998


Find the flux of the vector field **F** = (x - 2y + 1)i + (2x + y - 3z)j + (3y + z)k through the part of the surface of the sphere x² + y² + z² that lies in the first octant in the region where x² + y² + z² > 1.

Answer: \frac{3\pi}{4}


Let \( x \) be a complex number such that \( x + x^{-1} \) is a root of the polynomial \( p(t) = t^3 + t^2 - 2t - 1 \). Find all possible values of \( x^7 + x^{-7} \).

Answer: 2


By starting with a million and alternatively dividing by 2 and multiplying by 5, Anisha created a sequence of integers that starts 1000000, 500000, 2500000, 1250000, and so on. What is the last integer in her sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as possible.

Answer: 5^{12}


**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, what is the probability that she correctly dials Layla's phone number? Express the answer as a common fraction.**

Answer: \frac{1}{72}


Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator?

Answer: 11


Given $a\in R$, $b\in R$, if the set $A=\{a, \frac{b}{a}, 1\}$, $B=\{a^{2}, a+b, 0\}$, $A\subseteq B$, and $B\subseteq A$, then the value of $a^{2023}+b^{2023}$ is ______.

Answer: -1


Given three rays $AB$, $BC$, $BB_{1}$ are not coplanar, and the diagonals of quadrilaterals $BB_{1}A_{1}A$ and $BB_{1}C_{1}C$ bisect each other, and $\overrightarrow{AC_{1}}=x\overrightarrow{AB}+2y\overrightarrow{BC}+3z\overrightarrow{CC_{1}}$, find the value of $x+y+z$.

Answer: \frac{11}{6}


Car A and Car B start simultaneously from points $A$ and $B$ respectively, traveling towards each other. The initial speed ratio of car A to car B is 5:4. Shortly after departure, car A has a tire blowout, stops to replace the tire, and then resumes the journey, increasing its speed by $20\%$. They meet at the midpoint between $A$ and $B$ after 3 hours. After meeting, car B continues forward while car A turns back. When car A returns to point $A$ and car B reaches the position where car A had the tire blowout, how many minutes did car A spend replacing the tire?

Answer: 52


Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are denoted as $S_n$ and $T_n$ respectively. If for any natural number $n$, it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, calculate the value of $\dfrac{a_3+a_{15}}{2(b_3+b_9)}+ \dfrac{a_3}{b_2+b_{10}}$.

Answer: \dfrac{19}{41}


One night, 21 people exchanged phone calls $n$ times. It is known that among these people, there are $m$ people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ called $a_{i+1}$ (for $i=1,2, \cdots, m$ and $a_{m+1}=a_{1}$), and $m$ is an odd number. If no three people among these 21 people have all exchanged calls with each other, determine the maximum value of $n$.

Answer: 101


Given a triangle with sides  $a, b, c$  and  medians  $s_a, s_b, s_c$  respectively. Prove the following inequality:  $$ a + b + c> s_a + s_b + s_c> \frac34 (a + b + c)  $$

Answer: s_a+s_b+s_c>\frac{3}{4}()


Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.

Answer: \frac{1}{e}


Let the following system of equations be satisfied for positive numbers \(x, y, z\):

\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=147 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=163
\end{array}\right.
\]

Find the value of the expression \( x y+y z+x z \).

Answer: 56


Given \(\alpha, \beta \in [0, \pi]\), find the maximum value of \((\sin \alpha + \sin (\alpha + \beta)) \cdot \sin \beta\).

Answer: \frac{8\sqrt{3}}{9}


P, Q, R are three points in space. The circle \(C_P\) passes through Q and R, the circle \(C_Q\) passes through R and P, and the circle \(C_R\) passes through P and Q. The tangents to \(C_Q\) and \(C_R\) at P coincide. Similarly, the tangents to \(C_R\) and \(C_P\) at Q coincide, and the tangents to \(C_P\) and \(C_Q\) at R coincide. Show that the circles are either coplanar or lie on the surface of the same sphere.

Answer: Thecircleseithercoplanarorlieonthesurfaceofthesamesphere.


For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such  that the following two conditions are satisfied:
[list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$,
[*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list]
Determine $N(n)$ for all $n\geq 2$.

[i]

Answer: \left\lfloor \frac{2n}{3} \right\rfloor + 1


Given \( m \leq n \) are positive integers, and \( \binom{n}{m} \) denotes the binomial coefficient \( \frac{n!}{m! (n-m)!} \), show that \( \binom{n}{m} \binom{n}{m-1} \) is divisible by \( n \). Find the smallest positive integer \( k \) such that \( k \binom{n}{m} \binom{n}{m-1} \binom{n}{m-2} \) is divisible by \( n^2 \) for all \( m, n \) such that \( 1 < m \leq n \). For this value of \( k \) and fixed \( n \), find the greatest common divisor of the \( n - 1 \) integers:

\[
\frac{k \binom{n}{m} \binom{n}{m-1} \binom{n}{m-2}}{n^2}
\]

where \( 1 < m \leq n \).

Answer: n-1


Given that \( A \) and \( B \) are two subsets of \(\{1, 2, \ldots, 100\}\) such that \( |A| = |B| \), \( A \cap B = \emptyset \), and for any \( x \in A \), \( 2x + 2 \in B \). Find the maximum value of \( |A \cup B| \).

Answer: 66


Let $6n$ be a positive integer, and let $a_{j} (j=1, 2, \cdots, n)$ be complex numbers. For any non-empty subset $I$ of the set $\{1,2, \cdots, n\}$, it holds that
$$\left|\prod_{j \in I}\left(1+a_{j}\right)-1\right| \leqslant \frac{1}{2}.$$
Prove that:
$$\sum_{i=1}^{n}\left|a_{j}\right| \leqslant 3.$$

Answer: 3


Let $\mathbf{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -7 \\ 3 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$

Answer: 13


$\triangle{ABC}$  is isosceles $(AB=AC)$ . Points  $P$  and  $Q$  exist inside the triangle such that  $Q$  lies inside  $\widehat{PAC}$  and  $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$ . We also have  $BP=PQ=CQ$ .Let  $X$  and  $Y$  be the intersection points of  $(AP,BQ)$  and  $(AQ,CP)$  respectively. Prove that quadrilateral  $PQYX$  is cyclic. *(20 Points)*

Answer: PQYX


Janaína bought a different deck: there are 180 cards numbered from 1 to 180, with even cards being red and odd cards being blue.

a) How many cards are multiples of 7 and red?

b) How many cards are multiples of 7 or red?

c) Janaína randomly chose 80 cards, all of which, coincidentally, are black. What is the minimum number of cards that are multiples of 7?

Answer: 3


Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists.

Answer: 64


Bob chooses a  $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If  $N$  is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of  $N$ . For example, if Bob’s string is  $0000$  and the stream of bits begins  $101000001 \dots$ , then  $N = 7$ .

Answer: 30


In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\cos C - 2a - c = 0$.

(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Find the distance from the circumcenter of $\triangle ABC$ to side $AC$.

Answer: \frac{\sqrt{3}}{6}


In a Cartesian coordinate system, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\Omega: x^{2}-\frac{y^{2}}{3}=1$. A line $l$ passing through $F_{1}$ intersects $\Omega$ at two points $P$ and $Q$. If $\overrightarrow{F_{1} F_{2}} \cdot \overrightarrow{F_{1} P}=16$, find the value of $\overrightarrow{F_{2} P} \cdot \overrightarrow{F_{2} Q}$.

Answer: \frac{27}{13}


Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive?

Answer: 36


Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Answer: 1000


Determine the values of the real parameter  $a$ , such that the equation
\[\sin 2x\sin 4x-\sin x\sin 3x=a\]
has a unique solution in the interval  $[0,\pi)$ .

Answer: 1


Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour?
$\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 22$

Answer: \textbf{(E)}\22


Into each row of a \( 9 \times 9 \) grid, Nigel writes the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) in order, starting at one of the digits and returning to 1 after 9: for example, one row might contain \( 7, 8, 9, 1, 2, 3, 4, 5, 6 \). The grid is gorgeous if each nine-digit number read along a row or column or along the diagonal from the top-left corner to the bottom-right corner or the diagonal from the bottom-left corner to the top-right corner is divisible by 9. How many of the \( 9^{9} \) possible grids are gorgeous?

Answer: 9^8


Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$.

Answer: -4


Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
$\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$
$\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$
$\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$
$\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$
$\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

Answer: \textbf{(C)}


$A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $B D$ at $X . \omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D$.

Answer: \frac{115}{143}


Answer: 1005


Every quadratic equation can be reduced, by dividing it by the coefficient of the highest degree term, to the form

$$
x^{2}+c x+d=0
$$

The total number of distinct reduced quadratic equations in $p$-arithmetic equals $p^{2}$. Count how many of these equations have no roots, how many have one root, and how many have two distinct roots.

1) The expression $b^{2}-4 a c$ is called the discriminant of the equation $a x^{2}+b x+c=0$.

Answer: \frac{p(p-1)}{2}


Let $c$ and $d$ be real numbers such that
\[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\]
Find
\[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\]

Answer: \frac{9}{11}


Let $f_1(x) = \frac23 - \frac3{3x+1},$ and for $n \ge 2,$ define $f_n(x) = f_1(f_{n-1}(x)).$ Find the value of $x$ that satisfies $f_{1001}(x) = x-3.$

Answer: \tfrac{5}{3}


A cone is inscribed in a regular hexagonal pyramid, and another cone is circumscribed around it. Find the difference in volumes between the circumscribed cone and the inscribed cone if the height of the pyramid is \( H \) and the radius of the base of the circumscribed cone is \( R \).

Answer: \frac{1}{12}\piR^2H


A circle \(\omega\) is circumscribed around triangle \(ABC\). Tangents to the circle at points \(A\) and \(B\) intersect at point \(K\). Point \(M\) is the midpoint of side \(AC\). A line passing through point \(K\) and parallel to \(AC\) intersects side \(BC\) at point \(L\). Find the angle \(AML\).

Answer: 90


Let  $\omega$  and  $\Gamma$  be circles such that  $\omega$  is internally tangent to  $\Gamma$  at a point  $P$ . Let  $AB$  be a chord of  $\Gamma$  tangent to  $\omega$  at a point  $Q$ . Let  $R\neq P$  be the second intersection of line  $PQ$  with  $\Gamma$ . If the radius of  $\Gamma$  is  $17$ , the radius of  $\omega$  is  $7$ , and  $\frac{AQ}{BQ}=3$ , find the circumradius of triangle  $AQR$ .

Answer: \sqrt{170}


Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1.

Answer: 50


Integers from 1 to 100 are placed in a row in some order. Let us call a number *large-right*, if it is greater than each number to the right of it; let us call a number *large-left*, is it is greater than each number to the left of it. It appears that in the row there are exactly  $k$  large-right numbers and exactly  $k$  large-left numbers. Find the maximal possible value of  $k$ .

Answer: 50


Do there exist strictly positive integers \(a, b, c\) such that the four equations \(a x^{2} \pm b x \pm c=0\) each have two non-zero integer roots?

Answer: Yes


In the Kangaroo republic each month consists of 40 days, numbered 1 to 40. Any day whose number is divisible by 6 is a holiday, and any day whose number is a prime is a holiday. How many times in a month does a single working day occur between two holidays?

Answer: 1


From the 100 numbers $1, 2, 3, \cdots, 100$, select any 3 numbers. Find the probability that their sum is divisible by 3.

Answer: \frac{817}{2450}


We define \( F_n = 2^{2^n} + 1 \) for \( n \geq 2 \) and take a prime factor \( p \) of \( F_n \). Show that:
a) \( 2 \) is a quadratic residue modulo \( p \);
b) \( p \equiv 1 \pmod{2^{n+2}} \).

Answer: p\equiv1\pmod{2^{n+2}}


$2$  darts are thrown randomly at a circular board with center  $O$ , such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked  $A$  and  $B$ . What is the probability that  $\angle AOB$  is acute?

Answer: \frac{1}{2}


Let $n$ be the least positive integer greater than $1000$ for which
\[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.\]
What is the sum of the digits of $n$?
$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$

Answer: \textbf{(C)}18


Determine the number of all numbers which are represented as  $x^2+y^2$  with  $x, y \in \{1, 2, 3, \ldots, 1000\}$  and which are divisible by 121.

Answer: 8100


For a given positive integer $k$, find the smallest positive integer $N$ such that there exists a set consisting of $2k + 1$ distinct positive integers whose total sum is greater than $N$, but the sum of any $k$-element subset is at most $\frac{N}{2}$.

Answer: 2k^3+3k^2+3k


There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?

Answer: 20


The points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. A ray \( l_{1} \) is drawn from \( A_{1} \) in the direction of \( A_{2} \), a ray \( l_{2} \) is drawn from \( A_{2} \) in the direction of \( A_{3} \), and so on, with a ray \( l_{6} \) is drawn from \( A_{6} \) in the direction of \( A_{1} \). From a point \( B_{1} \) on the ray \( l_{1} \), a perpendicular is dropped onto the ray \( l_{6} \), from the foot of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The foot of the sixth perpendicular coincides with \( B_{1} \). Find the segment \( B_{1} A_{1} \).

Answer: 2


Through vertex $A$ of an isosceles acute-angled triangle $ABC$ and the center of the circumcircle around this triangle, a line is drawn intersecting side $BC$ at point $D$. Find the length of $AD$, given that $AB = BC = b$ and $\angle ABC = \alpha$.

Answer: \frac{b\sin\alpha}{\sin\frac{3\alpha}{2}}


Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$.  Enter all the solutions, separated by commas.

Answer: -\sqrt{3},-1,1,\sqrt{3}


Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself?

Answer: 14


Sam places a total of 30 yellow Easter eggs in several purple baskets and a total of 45 pink Easter eggs in some orange baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Sam put in each basket?

Answer: 15


How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit?

Answer: 715


If three numbers $a_1, a_2, a_3$ are selected from the set $\{1, 2, \cdots, 14\}$ in ascending order such that $a_2 - a_1 \geq 3$ and $a_3 - a_2 \geq 3$, how many different ways are there to select these numbers?

Answer: 120


Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \).

Answer: 25/3


Call a positive integer \( N \geq 2 \) "special" if for every \( k \) such that \( 2 \leq k \leq N \), \( N \) can be expressed as a sum of \( k \) positive integers that are relatively prime to \( N \) (although not necessarily relatively prime to each other). How many special integers are there less than 100?

Answer: 50


Given a prime number \( p \) and a natural number \( n \) such that \( p \geq n \geq 3 \). The set \( A \) consists of sequences of length \( n \) taken from the set \(\{1,2,\cdots,p-1\}\). For any two sequences \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) and \(\left(y_{1}, y_{2}, \cdots, y_{n}\right)\) in set \( A \), there exist three distinct positive integers \( k, l, m \) such that \( x_{k} \ne y_{k} \), \( x_{l} \ne y_{l} \), and \( x_{m} \ne y_{m} \). Find the maximum number of elements in set \( A \).

Answer: (p-1)^{n-2}


Three years ago, you invested some money at $12\%$ interest. You now have $\$504.32$ in the account. If the interest was compounded yearly, how much did you invest 3 years ago?

Answer: 359


In the cells of an  $8\times 8$  board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Answer: 36


When $1 - i \sqrt{3}$ is converted to the exponential form $re^{i \theta}$, what is $\theta$?

Answer: \frac{5\pi}{3}


Let  $\mathbb N$  be the set of positive integers. Determine all functions  $f:\mathbb N\times\mathbb N\to\mathbb N$  that satisfy both of the following conditions:

- $f(\gcd (a,b),c) = \gcd (a,f(c,b))$  for all  $a,b,c \in \mathbb{N}$ .
- $f(a,a) \geq a$  for all  $a \in \mathbb{N}$ .

Answer: f(,b)=\gcd(,b)


The gardener decided to create a new fruit orchard. He planted young trees in rows, forming a square. As a result, he had 146 extra saplings left. To expand the square by adding one more row, the gardener had to buy an additional 31 trees.

How many trees were there in the orchard at the end of the work?

Answer: 7921


If $y=x^2+px+q$, then if the least possible value of $y$ is zero $q$ is equal to:
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac{p^2}{4}\qquad\textbf{(C)}\ \frac{p}{2}\qquad\textbf{(D)}\ -\frac{p}{2}\qquad\textbf{(E)}\ \frac{p^2}{4}-q$

Answer: \textbf{(B)}\\frac{p^2}{4}


There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided?

Answer: 20


$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48.

Answer: 45


Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$?

Answer: 85


Given that complex numbers \( z_{1} \) and \( z_{2} \) satisfy \( \left(z_{1} - i\right)\left(z_{2} + i\right) = 1 \), and \( \left|z_{1}\right| = \sqrt{2} \), determine the range of values for \( \left|z_{2}\right| \).

Answer: [2-\sqrt{2},2+\sqrt{2}]


Two classes organize a joint club afternoon. To cover the costs, each student from class $A$ contributed $5 \mathrm{Ft}$, and each student from class $B$ contributed $3 \mathrm{Ft}$. If class $A$ students had covered all the costs, it would have cost each of them $K \mathrm{Ft}$. For the second joint event, class $A$ students contributed $4 \mathrm{Ft}$ each, and class $B$ students contributed $6 \mathrm{Ft}$ each, and the total amount of money collected was the same as if each student from class $B$ had paid $K$ \mathrm{Ft}. What is the value of $K$, and which class has more students?

Answer: 9


Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______.

Answer: -8


Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$.

Answer: 56


Let $\ell_A$ and $\ell_B$ be two distinct perpendicular lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than 1 of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8.0 regions when $m=3$ and $n=2$ [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]

Answer: 244


Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is:

Answer: 1


Given the function
\[ f(x) = \sqrt{2x^{2} + 2x + 41} - \sqrt{2x^{2} + 4x + 4} \quad (x \in \mathbb{R}), \]
determine the maximum value of \( f(x) \).

Answer: 5


Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. What is the time of arrival at point \( B \)?

Answer: 11:40


A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar?

Answer: 16


A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?

Answer: 300


Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive,
$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ]
$\textbf{(A)}\ 1 < x < 11\qquad \textbf{(B)}\ -1 < x < 11\qquad \textbf{(C)}\ x< 11\qquad \textbf{(D)}\ x>11\qquad \textbf{(E)}\ |x| < 6$

Answer: \textbf{(B)}\-1<x<11


Find all polynomials $R(x)$ of degree less than 4, for each of which there exists a polynomial $P(x)$ satisfying the identity $7 \sin^{31} t + 8 \sin^{18} t - 5 \sin^{5} t \cos^{4} t - 10 \sin^{2} t + 5 \sin^{5} t - 2 = P(\sin t) \left( \sin^{4} t - (1+\sin t) \left( \cos^{2} t - 2 \right) \right) + R(\sin t), \quad t \in \mathbb{R}$.

Answer: 13x^3+5x^2+12x+3


Using red, blue, and yellow paints to color the 6 faces of a cube so that each color appears on exactly 2 faces, find the number of distinct ways to color the cube.

Answer: 6


The insurance contract provides an insured amount of 500,000 rubles and a deductible of 1%. The actual damage amounted to 4,000 rubles. What will be the amount of insurance compensation?

Answer: 0


Given  $\triangle ABC$ , let  $R$  be its circumradius and  $q$  be the perimeter of its excentral triangle. Prove that  $q\le 6\sqrt{3} R$ .
Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.

Answer: q\le6\sqrt{3}R


For a positive integer  $n$ , let  $S_n$  be the total sum of the intervals of  $x$  such that  $\sin 4n x\geq \sin x$  in  $0\leq x\leq \frac{\pi}{2}.$ Find  $\lim_{n\to\infty} S_n.$ 60 point

Answer: \frac{\pi}{8}


A  $39$ -tuple of real numbers  $(x_1,x_2,\ldots x_{39})$  satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of  $\cos(x_1)$  and the maximum of  $\sin(x_1)$  over all tuples  $(x_1,x_2,\ldots x_{39})$  satisfying the condition is  $\tfrac ab$  for coprime positive integers  $a$ ,  $b$  (these maxima aren't necessarily achieved using the same tuple of real numbers). Find  $a + b$ .

*Proposed by Evan Chang*

Answer: 37


Given the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) with three points \(P\), \(Q\), and \(R\) on it, where \(P\) and \(Q\) are symmetric with respect to the origin. Find the maximum value of \(|RP| + |RQ|\).

Answer: 10


All integers from 1 to $2n$ are written in a row. Then the place number where each number is located is added to each number. Prove that among the resulting sums, there will be at least two that leave the same remainder when divided by $2n$.

Answer: 2


Given a regular decagon, the probability that exactly one of the sides of the triangle formed by connecting three randomly chosen vertices of the decagon is also a side of the decagon.

Answer: \frac{1}{2}


Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake.
Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?

Answer: 20


We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it?

Answer: \frac{\sqrt{5} - 1}{2}


Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.

(Ⅰ) If $\overrightarrow{PQ}=\lambda \overrightarrow{PR}$, find the value of $\lambda$;

(Ⅱ) If $\overrightarrow{SP} \perp \overrightarrow{PR}$, find the minimum value of the area of $\triangle PSR$.

Answer: \frac{4\sqrt{3}}{9}


In $\Delta ABC$, it is known that $c^2-a^2=5b$ and $3\sin A\cos C=\cos A\sin C$. Find the value of $b$.

Answer: 10


Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.

(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;

(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$.

Answer: 2-4\ln 2


Determine all triples  $(p, q, r)$  of positive integers, where  $p, q$  are also primes, such that  $\frac{r^2-5q^2}{p^2-1}=2$ .

Answer: (3,2,6)


If the radius of a circle is increased $100\%$, the area is increased:
$\textbf{(A)}\ 100\%\qquad\textbf{(B)}\ 200\%\qquad\textbf{(C)}\ 300\%\qquad\textbf{(D)}\ 400\%\qquad\textbf{(E)}\ \text{By none of these}$

Answer: \textbf{(C)}\300


Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\left(m-1\right)y+m-6=0$. Find the value of $m$.

Answer: -2


Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:
\[
\frac{1}{b^2 + c^2 - 3a^2} + \frac{1}{a^2 + c^2 - 3b^2} + \frac{1}{a^2 + b^2 - 3c^2}.
\]

Answer: -3


Given that $ab= \frac{1}{4}$, $a$, $b \in (0,1)$, find the minimum value of $\frac{1}{1-a}+ \frac{2}{1-b}$.

Answer: 4+ \frac{4 \sqrt{2}}{3}


Given an isosceles trapezoid with \(AB = 24\) units, \(CD = 10\) units, and legs \(AD\) and \(BC\) each measuring \(13\) units. Find the length of diagonal \(AC\).

Answer: 13


Quadrilateral \(ABCD\) is inscribed in a circle. Point \(X\) lies on its side \(AD\), with \(BX \parallel CD\) and \(CX \parallel BA\). Find \(BC\) if \(AX = \frac{3}{2}\) and \(DX = 6\).

Answer: 3


Given set \( S \) such that \( |S| = 10 \). Let \( A_1, A_2, \cdots, A_k \) be non-empty subsets of \( S \), and the intersection of any two subsets contains at most two elements. Find the maximum value of \( k \).

Answer: 175


Given non-zero vectors $a=(-x, x)$ and $b=(2x+3, 1)$, where $x \in \mathbb{R}$.

$(1)$ If $a \perp b$, find the value of $x$;

$(2)$ If $a \nparallel b$, find $|a - b|$.

Answer: 3\sqrt{2}


Show that if \( x \) is the measure of an acute angle, then

\[ \sin(\cos x) < \cos(\sin x). \]

Answer: \sin(\cosx)<\cos(\sinx)


The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________.

Answer: 2\pi


In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \le n \le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original starting position after all 30 moves?

Answer: 37


This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant  $X$ , let  $t(X)$  be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$  For any two friends  $X'$  and  $X''$ , we have  $t(X') \neq t(X''),$  $2)$  For every contestant  $X$ , the set  $\{ t(Y) : Y \text{ is a friend of } X \}$  consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

Answer: 2


Let  $p\ge 3$  be a prime number. Each side of a triangle is divided into  $p$  equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.

Answer: 3p^2-3p+1


Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.

Answer: 72


An organization initially consists of 10 leaders and a certain number of regular members. Each year, all leaders are replaced, and each regular member recruits two new members. After this, 10 new leaders are elected from outside the organization. Initially, there are 30 people in total in the organization. How many people will be in the organization after 10 years?

Answer: 1180990


The distances from point \( M \), located inside triangle \( ABC \), to its sides \( AC \) and \( BC \) are 2 and 4, respectively. Find the distance from point \( M \) to line \( AB \), given that \( AB = 10 \), \( BC = 17 \), and \( AC = 21 \).

Answer: \frac{29}{5}


Given positive real numbers \( a, b, c \) that satisfy:
\[
a^{2} = b^{2} + bc \quad \text{and} \quad b^{2} = c^{2} + ac,
\]
prove that \( \frac{1}{c} = \frac{1}{a} + \frac{1}{b} \).

Answer: \frac{1}{}=\frac{1}{}+\frac{1}{b}


Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.

Answer: \frac{2}{3}


How many different routes can Samantha take by biking on streets to the southwest corner of City Park, then taking a diagonal path through the park to the northeast corner, and then biking on streets to school?

Answer: 400


Through two vertices of an equilateral triangle \(ABC\) with an area of \(21 \sqrt{3} \ \text{cm}^2\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle.

Answer: 2\sqrt{7}


Junior and Carlson ate a barrel of jam and a basket of cookies, starting and finishing at the same time. Initially, Junior ate the cookies and Carlson ate the jam, then (at some point) they switched. Carlson ate both the jam and the cookies three times faster than Junior. What fraction of the jam did Carlson eat, given that they ate an equal amount of cookies?

Answer: 9/10


Xiao Ming's family has three hens. The first hen lays one egg every day, the second hen lays one egg every two days, and the third hen lays one egg every three days. Given that all three hens laid eggs on January 1st, how many eggs did these three hens lay in total in the 31 days of January?

Answer: 56


Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \in \mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality
\[
\frac{1}{x-20}+\frac{1}{x-17} \geqslant \frac{1}{512}
\]

Answer: 1024


Color three faces of an $8 \times 8 \times 8$ cube red and three faces blue (ensuring that any three faces sharing the same vertex are not all the same color), and then cut it into 512 smaller $1 \times 1 \times 1$ cubes. How many of these 512 smaller cubes have at least one red face and one blue face?

Answer: 56


If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another example is the four-digit number $5324$, since $53-32=21\neq 24$, is not a "decreasing number". If a "decreasing number" is $\overline{a312}$, then this number is ______; if the sum of the three-digit number $\overline{abc}$ formed by the first three digits and the three-digit number $\overline{bcd}$ formed by the last three digits of a "decreasing number" is divisible by $9$, then the maximum value of the number that satisfies the condition is ______.

Answer: 8165


Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$

If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$

then the smallest positive period of \( f(x) \) is ______.

Answer: \pi


Let  $\alpha,\beta$  be the distinct positive roots of the equation of  $2x=\tan x$ .
Evaluate

\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

Answer: 0


\( S(n) \) represents the sum of the digits of the natural number \( n \), for example, \( S(123) = 1 + 2 + 3 = 6 \). If two different positive integers \( m \) and \( n \) satisfy
\[
\left\{\begin{array}{l}
m < 100 \\
n < 100 \\
m > n \\
m + S(n) = n + 2S(m)
\end{array}, \text{ then we call } m \text{ and } n \text{ a pair } \langle m, n \rangle.
\right.
\]

How many such pairs \( \langle m, n \rangle \) exist?

Answer: 99


Given the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.
(1) Find the equation of ellipse C;
(2) Let $F_1$, $F_2$ be the left and right foci of ellipse C, respectively. A line l that does not pass through $F_1$ intersects ellipse C at two distinct points A and B. If the slopes of lines $AF_1$ and $BF_1$ form an arithmetic sequence, find the maximum area of △AOB.

Answer: \sqrt{3}


In a certain state, license plate numbers are composed of 6 digits (from the digits 0-9). The state mandates that any two license plate numbers must differ by at least two digits (for example, 027592 and 020592 cannot both be used). Determine the maximum number of distinct license plate numbers possible, and provide a proof. (19th US Mathematical Olympiad, 1990)

Answer: 100000


For every natural number  $n\geq{2}$  consider the following affirmation  $P_n$ :
"Consider a polynomial  $P(X)$  (of degree  $n$ ) with real coefficients. If its derivative  $P'(X)$  has  $n-1$  distinct real roots, then there is a real number  $C$  such that the equation  $P(x)=C$  has  $n$  real,distinct roots."
Are  $P_4$  and  $P_5$  both true? Justify your answer.

Answer: P_4


Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:
- Less than 1 year: 3 marks
- 1 to less than 2 years: 6 marks
- 2 to less than 3 years: 5 marks
- 3 to less than 4 years: 4 marks
- 4 to less than 5 years: 2 marks
- 5 to less than 6 years: 2 marks
- 6 to less than 7 years: 3 marks
- 7 to less than 8 years: 2 marks
- 8 to less than 9 years: 1 mark
- 9 to less than 10 years: 1 mark
Determine what percent of the employees have worked there for $6$ years or more.

Answer: 24.14\%


The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$.

Answer: 21


A tourist city was surveyed, and it was found that the number of tourists per day $f(t)$ (in ten thousand people) and the time $t$ (in days) within the past month (calculated as $30$ days) approximately satisfy the function relationship $f(t)=4+ \frac {1}{t}$. The average consumption per person $g(t)$ (in yuan) and the time $t$ (in days) approximately satisfy the function relationship $g(t)=115-|t-15|$.

(I) Find the function relationship of the daily tourism income $w(t)$ (in ten thousand yuan) and time $t(1\leqslant t\leqslant 30,t\in N)$ of this city;

(II) Find the minimum value of the daily tourism income of this city (in ten thousand yuan).

Answer: 403 \frac {1}{3}


Pentagon \(ABCDE\) is inscribed in a circle. The distances from point \(A\) to the lines \(BC\), \(CD\), and \(DE\) are \(a\), \(b\), and \(c\) respectively.

Find the distance from vertex \(A\) to the line \(BE\).

Answer: \frac{ac}{b}


In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are there in the hospital?

Answer: 20


Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that
$DF=\frac{1}{3}DA$. What is the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$?
$\textbf{(A)}\ 1:2 \qquad \textbf{(B)}\ 1:3 \qquad \textbf{(C)}\ 1:5 \qquad \textbf{(D)}\ 1:6 \qquad \textbf{(E)}\ 1:7$

Answer: \textbf{(C)}\1:5


There is an unlimited supply of square glass panes in 10 colors. How many ways are there to insert 4 panes into a $2 \times 2$ window frame such that some color appears in both the top and bottom halves of the window?

Answer: 3430


Let \( f(n) \) be the integer nearest to \( \sqrt{n} \). Find the value of
\[
\sum_{n=1}^{\infty} \frac{\left(\frac{3}{2}\right)^{f(n)} + \left(\frac{3}{2}\right)^{-f(n)}}{\left(\frac{3}{2}\right)^{n}}
\]

Answer: 5


A circle is tangent to the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $\sqrt{2+\sqrt{2}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $45^{\circ}$, and knowing that $\sin 22.5^{\circ} = \frac{\sqrt{2-\sqrt{2}}}{2}$.

Answer: \sqrt{2}+\sqrt{2-\sqrt{2}}


In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is

Answer: 48


From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?

(2004 National Girls' Olympiad problem)

Answer: 1006009


A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \text{ cm}\) to a cylinder of radius \(3 \text{ cm}\). When the bottle is right side up, the height of the water inside is \(20 \text{ cm}\). When the bottle is upside down, the height of the liquid is \(28 \text{ cm}\). What is the total height, in \(\text{cm}\), of the bottle?

Answer: 29


On the sides \(AB\) and \(AC\) of an acute-angled triangle \(ABC\), points \(K\) and \(L\) are marked such that the quadrilateral \(BKLC\) is inscribed. Inside this quadrilateral, a point \(M\) is chosen such that the line \(AM\) is the bisector of angle \(BMC\). Ray \(BM\) intersects the circumcircle of triangle \(AMC\) at point \(P\) for the second time, and ray \(CM\) intersects the circumcircle of triangle \(AMB\) at point \(Q\) for the second time. Find the ratio of the areas of triangles \(ALP\) and \(AKQ\).

Answer: 1


Let  $ABC$  be a triangle such that  $AB=6,BC=5,AC=7.$  Let the tangents to the circumcircle of  $ABC$  at  $B$  and  $C$  meet at  $X.$  Let  $Z$  be a point on the circumcircle of  $ABC.$  Let  $Y$  be the foot of the perpendicular from  $X$  to  $CZ.$  Let  $K$  be the intersection of the circumcircle of  $BCY$  with line  $AB.$  Given that  $Y$  is on the interior of segment  $CZ$  and  $YZ=3CY,$  compute  $AK.$

Answer: 147/10


On each of the lines \( x=2 \) and \( x=9 \), there are 400 points marked with ordinates \( 1, 2, 3, \ldots, 400 \). In how many ways can three points be chosen from the 800 marked points so that they form the vertices of a right triangle?

Answer: 321372


In an acute-angled triangle \( \triangle ABC \), \( M \) is the midpoint of side \( AC \). A circle \(\Gamma\) passing through points \( B \) and \( M \) intersects sides \( AB \) and \( BC \) at points \( P \) and \( Q \) respectively, besides point \( B \). Point \( T \) satisfies that quadrilateral \( BPQT \) is a parallelogram. If \( T \) is on the circumcircle of \( \triangle ABC \), find all possible values of \( \frac{BT}{BM} \).

Answer: \sqrt{2}


Let \( p, q, r, s \) be the four roots of the equation \( 2(10x + 13)^2 (5x + 8)(x + 1) = 1 \). If \( pq + rs \) is real, find the value of this real number.

Answer: \frac{329}{100}


Given \(x, y, z \in (0, 2]\), find the maximum value of the expression

\[
A = \frac{\left(x^{3} - 6\right) \sqrt[3]{x + 6} + \left(y^{3} - 6\right) \sqrt[3]{y + 6} + \left(z^{3} - 6\right) \sqrt[3]{z + 6}}{x^{2} + y^{2} + z^{2}}
\]

Answer: 1


The height of the triangular pyramid \(ABCD\) dropped from vertex \(D\) passes through the point of intersection of the heights of triangle \(ABC\). Additionally, it is known that \(DB = 3\), \(DC = 2\), and \(\angle BDC = 90^{\circ}\). Find the ratio of the area of face \(ADB\) to the area of face \(ADC\).

Answer: \frac{3}{2}


Let $P$, $Q$, and $R$ be points on a circle of radius $24$. If $\angle PRQ = 40^\circ$, what is the circumference of the minor arc $PQ$? Express your answer in terms of $\pi$.

Answer: \frac{32\pi}{3}


A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$?

Answer: $6$


Prove by computation that the volume of the solid obtained by rotating a circular segment with chord \( a \) about a diameter parallel to this chord does not depend on the radius of the circle.

Answer: \frac{\pi^3}{6}


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that

\[ f(x+y) = x + f(f(y)) \]

for all real numbers \( x \) and \( y \).

Answer: f(x)=x


The vertex $A$ of parallelogram $A B C D$ is connected by segments with the midpoints of the sides $B C$ and $C D$. One of these segments turned out to be twice as long as the other. Determine whether angle $B A D$ is acute, right, or obtuse.

Answer: Obtuse


If two 4'' by 4'' squares are added at each successive stage, what will be the area of the rectangle at Stage 4, in square inches?

Answer: 128


Find all values of \( x \) for which the minimum of the numbers \( 8 - x^{2} \) and \( \operatorname{ctg} x \) is not less than -1. In the answer, record the total length of the found intervals on the number line, rounding it to the nearest hundredth if necessary.

Answer: 4.57


How many ten digit positive integers with distinct digits are multiples of  $11111$ ?

Answer: 3456


Find all strictly increasing functions  $f : \mathbb{N} \to \mathbb{N} $  such that  $\frac {f(x) + f(y)}{1 + f(x + y)}$  is a non-zero natural number, for all  $x, y\in\mathbb{N}$ .

Answer: f(x)=ax+1


Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \).

Answer: 31


Determine the number of ways of walking from (0,0) to (5,5) using only up and right unit steps such that the path does not pass through any of the following points: (1,1), (1,4), (4,1), (4,4).

Answer: 34


A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period?

Answer: 43


A line moves such that it always remains parallel to a given plane $\alpha$ and at the same time intersects two given skew lines $a$ and $b$. What path does a point on such a line describe if it divides the segment of this line between its intersections with lines $a$ and $b$ in a constant ratio?

Answer: straightline


The function \( f \) is defined on the natural numbers \( 1, 2, 3, \ldots \) by \( f(1)=1 \) and
\[
f(n)=\left\{\begin{array}{ll}
f\left(\frac{n}{10}\right) & \text{if } 10 \mid n \\
f(n-1)+1 & \text{otherwise}
\end{array}\right.
\]

(a) Calculate \( f(2019) \).
(b) Determine the maximum value of \( f(n) \) for \( n \leq 2019 \).
(c) A new function \( g \) is defined by \( g(1)=1 \) and
\[
g(n)=\left\{\begin{array}{ll}
g\left(\frac{n}{3}\right) & \text{if } 3 \mid n \\
g(n-1)+1 & \text{otherwise}
\end{array}\right.
\]

Determine the maximum value of \( g(n) \) for \( n \leq 100 \).

Answer: 8


Given the parametric equations \(\left\{\begin{array}{l}x=\frac{a t}{1+t^{2}}, \\ y=\frac{1-t^{2}}{1+t^{2}},\end{array}\right.\) where \(a \in \mathbf{R}^{\prime}\), determine what type of curve they represent.

Answer: ellipse


In a certain country, there are 21 cities, and the government intends to build $n$ roads (all two-way), with each road connecting exactly 2 of the country’s cities. What is the minimum value of $n$ such that, regardless of how the roads are built, it is possible to travel between any 2 cities (possibly passing through intermediate cities)?

Answer: 191


Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$ with the sums of the first $n$ terms being $S_{n}$ and $T_{n}$, respectively, if $\frac{S_n}{T_n}=\frac{3n+4}{n+2}$, find $\frac{a_3+a_7+a_8}{b_2+b_{10}}$.

Answer: \frac{111}{26}


Let \( G_1 G_2 G_3 \) be a triangle with \( G_1 G_2 = 7 \), \( G_2 G_3 = 13 \), and \( G_3 G_1 = 15 \). Let \( G_4 \) be a point outside triangle \( G_1 G_2 G_3 \) so that ray \( \overrightarrow{G_1 G_4} \) cuts through the interior of the triangle, \( G_3 G_4 = G_4 G_2 \), and \( \angle G_3 G_1 G_4 = 30^\circ \). Let \( G_3 G_4 \) and \( G_1 G_2 \) meet at \( G_5 \). Determine the length of segment \( G_2 G_5 \).

Answer: \frac{169}{23}


Evaluate the value of the expression \((5(5(5(5+1)+1)+1)+1)\).

Answer: 781


Emilia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions. Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?

Answer: 1


Let  $S=\{1,2,\ldots ,98\}$ . Find the least natural number  $n$  such that we can pick out  $10$  numbers in any  $n$ -element subset of  $S$  satisfying the following condition: no matter how we equally divide the  $10$  numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

Answer: 50


Given a semicircle with diameter $A B$ and center $O$, and a line intersecting the semicircle at points $C$ and $D$, and the line $A B$ at point $M$ (with $MB < MA$ and $MD < MC$). Let $K$ be the point, different from $O$, where the circumcircles of triangles $AOC$ and $DOB$ intersect. Prove that angle $MKO$ is a right angle.

Answer: \angleMKO=90


Given that the numbers - 2, 5, 8, 11, and 14 are arranged in a specific cross-like structure, find the maximum possible sum for the numbers in either the row or the column.

Answer: 36


A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number.

Answer: 120


The circles $\alpha, \beta, \gamma$, and $\delta$ are tangent to a given circle at the vertices $A, B, C,$ and $D$ of a convex quadrilateral $ABCD$. Let $t_{\alpha \beta}$ be the length of the common tangent to the circles $\alpha$ and $\beta$ (external if both tangencies are internal or both are external at the same time, and internal if one tangency is internal and the other is external); $t_{\beta \gamma}, t_{\gamma \delta}$, etc., are defined similarly. Prove that $t_{\alpha \beta} t_{\gamma \delta} + t_{\beta \gamma} t_{\delta \alpha} = t_{\alpha \gamma} t_{\beta \delta}$ (generalized Ptolemy's theorem).

Answer: t_{\alpha\beta}t_{\gamma\delta}+t_{\beta\gamma}t_{\delta\alpha}=t_{\alpha\gamma}t_{\beta\delta}


If the system of inequalities with respect to $x$:
\[
\left\{
\begin{array}{l}
x^{3} + 3x^{2} - x - 3 > 0 \\
x^{2} - 2ax - 1 \leqslant 0
\end{array}
\right., \quad (a > 0)
\]
has exactly one integer solution, then determine the range of values for $a$.

Answer: [\frac{3}{4},\frac{4}{3})


A *$n$ -trønder walk* is a walk starting at  $(0, 0)$ , ending at  $(2n, 0)$  with no self intersection and not leaving the first quadrant, where every step is one of the vectors  $(1, 1)$ ,  $(1, -1)$  or  $(-1, 1)$ . Find the number of  $n$ -trønder walks.

Answer: (2n-1)!!


Given ten points on the boundary line of a half-plane, how many ways can the points be paired such that the points paired together can be connected by non-intersecting lines within the half-plane?

Answer: 42


Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble among any number of marbles. He charges 1 forint per measurement, but if there is a radioactive marble among those being measured, all of the marbles in the measurement will become radioactive.

What is the maximum profit I can absolutely achieve?

Answer: 92


Let  $ABC$  be an equilateral triangle with side length  $1$ . Points  $A_1$  and  $A_2$  are chosen on side  $BC$ , points  $B_1$  and  $B_2$  are chosen on side  $CA$ , and points  $C_1$  and  $C_2$  are chosen on side  $AB$  such that  $BA_1<BA_2$ ,  $CB_1<CB_2$ , and  $AC_1<AC_2$ .
Suppose that the three line segments  $B_1C_2$ ,  $C_1A_2$ ,  $A_1B_2$  are concurrent, and the perimeters of triangles  $AB_2C_1$ ,  $BC_2A_1$ , and  $CA_2B_1$  are all equal. Find all possible values of this common perimeter.

*Ankan Bhattacharya*

Answer: 1


Suppose the real number \( a \) satisfies \( a < 9a^3 - 11a < |a| \). What is the range of values of \( a \)?

Answer: (-\frac{2\sqrt{3}}{3},-\frac{\sqrt{10}}{3})


Find all pairs of integers $a$ and $b$ for which
\[7a+14b=5a^2+5ab+5b^2\]

Answer: (0,0),(1,2),(-1,3)


Given point $P(0,-2)$, and an ellipse $E$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$ with eccentricity $\dfrac{\sqrt{2}}{2}$, let $F$ be the right focus of ellipse $E$, and the slope of line $PF$ is $2$, with $O$ being the origin.
$(1)$ Find the equation of ellipse $E$;
$(2)$ A line $l$ intersects the circle $O$: $x^{2}+y^{2}=3$ and forms a chord of length $3$, and intersects ellipse $E$ at points $A$ and $B$. Find the maximum area of $\triangle AOB$.

Answer: \dfrac{\sqrt{2}}{2}


What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?

Answer: 15


Let  $a, b, c$  be positive integers such that  $a + 2b +3c = 100$ .
Find the greatest value of  $M = abc$

Answer: 6171


Let \( a \), \( b \), and \( c \) be the side lengths of a triangle, and assume that \( a \leq b \) and \( a \leq c \). Let \( x = \frac{b + c - a}{2} \). If \( r \) and \( R \) denote the inradius and circumradius, respectively, find the minimum value of \( \frac{a x}{r R} \).

Answer: 3


By definition, \( n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n \). Which factor should be crossed out from the product \( 1! \cdot 2! \cdot 3! \cdot \ldots \cdot 20! \) so that the remaining product becomes the square of some natural number?

Answer: 10!


Let  $a,b,n$  positive integers with  $a>b$  and  $ab-1=n^2$ . Prove that  $a-b \geq \sqrt{4n-3}$  and study the cases where the equality holds.

Answer: b\ge\sqrt{4n-3}


How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6?

Answer: 11


Find the smallest value of  $n$  for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds  $3^{2007}$ .

Answer: 2000


On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identification of a box that contains a white ball?

Answer: 2003


Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).

$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$

Answer: \frac{5\pi}{6}


The four complex roots of
\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0,\]when plotted in the complex plane, form a rhombus.  Find the area of the rhombus.

Answer: \sqrt{10}


Circles \( C \) and \( C' \) intersect at \( O \) and \( X \). A circle with center \( O \) meets \( C \) at \( Q \) and \( R \) and meets \( C' \) at \( P \) and \( S \). \( PR \) and \( QS \) meet at \( Y \) distinct from \( X \). Show that \(\angle YXO = 90^\circ \).

Answer: 90


An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:

- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.

Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

Answer: 512


On semicircle, with diameter  $|AB|=d$ , are given points  $C$  and  $D$  such that:  $|BC|=|CD|=a$  and  $|DA|=b$  where  $a, b, d$  are different positive integers. Find minimum possible value of  $d$

Answer: 8


In Sichuan, the new college entrance examination was launched in 2022, and the first new college entrance examination will be implemented in 2025. The new college entrance examination adopts the "$3+1+2$" mode. "$3$" refers to the three national unified examination subjects of Chinese, Mathematics, and Foreign Languages, regardless of whether they are arts or sciences; "$1$" refers to choosing one subject from Physics and History; "$2$" refers to choosing two subjects from Political Science, Geography, Chemistry, and Biology. The subject selection situation of the first-year high school students in a certain school in 2022 is shown in the table below:

| Subject Combination | Physics, Chemistry, Biology | Physics, Chemistry, Political Science | Physics, Chemistry, Geography | History, Political Science, Geography | History, Political Science, Biology | History, Chemistry, Political Science | Total |
|---------------------|-----------------------------|-------------------------------------|-----------------------------|--------------------------------------|----------------------------------|--------------------------------------|-------|
| Male                | 180                         | 80                                  | 40                          | 90                                   | 30                               | 20                                   | 440   |
| Female              | 150                         | 70                                  | 60                          | 120                                  | 40                               | 20                                   | 460   |
| Total               | 330                         | 150                                 | 100                         | 210                                  | 70                               | 40                                   | 900   |

$(1)$ Complete the $2\times 2$ contingency table below and determine if there is a $99\%$ certainty that "choosing Physics is related to the gender of the student":

|                   | Choose Physics | Do not choose Physics | Total |
|-------------------|----------------|-----------------------|-------|
| Male              |                |                       |       |
| Female            |                |                       |       |
| Total             |                |                       |       |

$(2)$ From the female students who chose the combinations of History, Political Science, Biology and History, Chemistry, Political Science, select 6 students using stratified sampling to participate in a history knowledge competition. Find the probability that the 2 selected female students are from the same combination.

Given table and formula: ${K^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$

| $P(K^{2}\geqslant k_{0})$ | 0.15 | 0.1  | 0.05 | 0.01 |
|---------------------------|------|------|------|------|
| $k_{0}$                   | 2.072| 2.706| 3.841| 6.635|

Answer: \frac{7}{15}


There are \(n\) girls \(G_{1}, \ldots, G_{n}\) and \(n\) boys \(B_{1}, \ldots, B_{n}\). A pair \((G_{i}, B_{j})\) is called suitable if and only if girl \(G_{i}\) is willing to marry boy \(B_{j}\). Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?

Answer: \frac{n(n+1)}{2}


Compute the positive difference between the two real solutions to the equation   $$ (x-1)(x-4)(x-2)(x-8)(x-5)(x-7)+48\sqrt 3 = 0. $$

Answer: \sqrt{25+8\sqrt{3}}


The arithmetic mean (average) of the first $n$ positive integers is:
$\textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2}$

Answer: \textbf{(E)}\\frac{n+1}{2}


Answer: 2\\sqrt{3}


A  $3$  by  $3$  determinant has three entries equal to  $2$ , three entries equal to  $5$ , and three entries equal to  $8$ . Find the maximum possible value of the determinant.

Answer: 405


Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered?
$\textbf{(A)} \ 71 \qquad \textbf{(B)} \ 76 \qquad \textbf{(C)} \ 80 \qquad \textbf{(D)} \ 82 \qquad \textbf{(E)} \ 91$

Answer: \textbf{(C)}\80


Answer: -4


Answer: 41


In the triangle \( ABC \), the bisectors of \(\angle A\) and \(\angle B\) meet at the incentre \( I \). The extension of \( AI \) meets the circumcircle of triangle \( ABC \) at \( D \). Let \( P \) be the foot of the perpendicular from \( B \) onto \( AD \), and \( Q \) a point on the extension of \( AD \) such that \( ID = DQ \). Determine the value of \( \frac{BQ \times IB}{BP \times ID} \).

Answer: 2


Prove that the roots of any quadratic equation \( ax^2 + bx + c = 0 \) (with \( a, b \neq 0 \)) satisfy the inequality

\[ |x| \leqq \frac{2|ac| + b^2}{|ab|}. \]

Answer: |x|\le\frac{2|ac|+b^2}{||}


Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$.

Answer: \frac{5\pi}{12}


Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--10dir(180),linewidth(2));
draw((0,0)--10dir(67.5),linewidth(2));
draw((0,0)--10dir(0),dashed);

draw(10dir(180)..10dir(90)..10dir(0),linewidth(2));

draw((5,0)+5dir(180)..(5,0)+5dir(-90)..(5,0)+5dir(0),linewidth(2));

dot((0,0));
dot((5,0));

label("A",10dir(180),W);
label("B",10dir(0),E);
label("C",(0,0),SW);
label("D",5dir(0),NE);
label("P",10dir(67.5),NE);
[/asy]

Answer: 112.5


Given that $\alpha$ and $\beta$ are acute angles, $\cos\alpha=\frac{{\sqrt{5}}}{5}$, $\cos({\alpha-\beta})=\frac{3\sqrt{10}}{10}$, find the value of $\cos \beta$.

Answer: \frac{\sqrt{2}}{10}


In a right square prism $A B C D A_{1} B_{1} C_{1} D_{1}$, the height is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$.

Answer: \frac{\pi}{2}


Let \( P, Q, R, \) and \( S \) be positive real numbers such that
\[
\log_{10} (PQ) + \log_{10} (PS) = 2, \\
\log_{10} (QS) + \log_{10} (QR) = 3, \\
\log_{10} (RP) + \log_{10} (RS) = 5.
\]
Compute the value of the product \( PQRS \).

Answer: 100000


We placed a cube on the center of a potter's wheel, with a natural number written on each of its faces. Just before we spun the wheel, from our point of view, we could see three faces of the cube with numbers totaling 42. After rotating the potter's wheel by $90^{\circ}$, we observed three faces with numbers totaling 34, and after another $90^{\circ}$ rotation, we still saw three numbers totaling 53.

1. Determine the sum of the three numbers we will see from our point of view if the wheel is rotated another $90^{\circ}$.
2. The cube always rested on the face with the number 6. Determine the maximum possible sum of all six numbers on the cube.

Answer: 100


Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible values of \(a\), given that \(a + b = 114.5\).

Answer: \frac{190}{3}


In triangle \( \triangle ABC \), points \( P_1 \) and \( P_2 \) lie on side \( AB \) such that point \( P_2 \) is on segment \( BP_1 \) and \( AP_1 = BP_2 \). Points \( Q_1 \) and \( Q_2 \) lie on side \( BC \) such that point \( Q_2 \) is on segment \( BQ_1 \) and \( BQ_1 = CQ_2 \). Denote the intersection of line segments \( P_1 Q_2 \) and \( P_2 Q_1 \) as \( R \). Let the second intersection point of the circumcircles of \( \triangle P_1 P_2 R \) and \( \triangle Q_1 Q_2 R \) be \( S \), with \( S \) located inside \( \triangle P_1 Q_1 R \). If \( M \) is the midpoint of \( AC \), prove:
$$
\angle P_1 R S = \angle Q_1 R M.
$$

Answer: \angleP_1RS=\angleQ_1RM


Find all real numbers  $a$  for which there exists a non-constant function  $f :\Bbb R \to \Bbb R$  satisfying the following two equations for all  $x\in \Bbb R:$ i)  $f(ax) = a^2f(x)$  and
ii)  $f(f(x)) = a f(x).$

Answer: 0,1


Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\)

Answer: 103


Find \(\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)\) in terms of \(n\).

Answer: (-1)^{n-1}


Place \( n \) (\( n \geq 1 \)) distinct balls into 4 distinct boxes \( A_1, A_2, A_3, A_4 \) such that \( A_1 \) contains an odd number of balls and \( A_2 \) contains an even number of balls. Find the number of different methods \( g_n \) to distribute the balls.

Answer: 4^{n-1}


Anya calls a date beautiful if all 6 digits in its representation are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023?

Answer: 30


Given two parabolas $N\_1$: $y=ax^{2}+bx+c$ and $N\_2$: $y=-ax^{2}+dx+e$ with vertices $P\_1(x\_1,y\_1)$ and $P\_2(x\_2,y\_2)$, respectively. The parabolas intersect at points $A(12,21)$ and $B(28,3)$ (both distinct from the vertices). Determine the value of $\frac{x\_1+x\_2}{y\_1+y\_2}$.

Answer: \frac{5}{3}


 $a,b,c$  - are sides of triangle  $T$ . It is known, that if we increase any one  side by  $1$ , we get new
a) triangle
b)acute triangle

Find minimal possible area of triangle  $T$  in case of a) and in case b)

Answer: \frac{\sqrt{3}}{4}


Let  $S$  be a subset of the natural numbers such that  $0\in S$ , and for all  $n\in\mathbb N$ , if  $n$  is in  $S$ , then both  $2n+1$  and  $3n+2$  are in  $S$ .  What is the smallest number of elements  $S$  can have in the range  $\{0,1,\ldots, 2019\}$ ?

Answer: 47


Given a cyclic quadrilateral \(ABCD\). The rays \(AB\) and \(DC\) intersect at point \(K\). It turns out that points \(B\), \(D\), and the midpoints of segments \(AC\) and \(KC\) lie on the same circle. What values can the angle \(ADC\) take?

Answer: 90


Find the largest root of the equation
$$
3 \sqrt{x-2} + 2 \sqrt{2x+3} + \sqrt{x+1} = 11
$$

Answer: 3


Divide the sequence successively into groups with the first parenthesis containing one number, the second parenthesis two numbers, the third parenthesis three numbers, the fourth parenthesis four numbers, the fifth parenthesis one number, and so on in a cycle: $(3)$, $(5,7)$, $(9,11,13)$, $(15,17,19,21)$, $(23)$, $(25,27)$, $(29,31,33)$, $(35,37,39,41)$, $(43)$, $…$, then calculate the sum of the numbers in the 104th parenthesis.

Answer: 2072


How many multiples of 5 are between 100 and 400?

Answer: 59