Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finished?

Answer: 120


Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is $AC$?

Answer: 6


Let  $\triangle ABC$  be equilateral with integer side length. Point  $X$  lies on  $\overline{BC}$  strictly between  $B$  and  $C$  such that  $BX<CX$ . Let  $C'$  denote the reflection of  $C$  over the midpoint of  $\overline{AX}$ . If  $BC'=30$ , find the sum of all possible side lengths of  $\triangle ABC$ .

*Proposed by Connor Gordon*

Answer: 130


You walk for 90 minutes at a rate of 3 mph, then rest for 15 minutes, and then cycle for 45 minutes at a rate of 20 kph. Calculate the total distance traveled in 2 hours and 30 minutes.

Answer: 13.82


(Geometry Proof Exercise) From a point A outside a circle ⊙O with radius 2, draw a line intersecting ⊙O at points C and D. A line segment AB is tangent to ⊙O at B. Given that AC=4 and AB=$4 \sqrt {2}$, find $tan∠DAB$.

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


Suppose a family of sets $\mathscr{F}$ satisfies: for any three sets
$$
\begin{array}{l}
X_{1} \backslash X_{2}, X_{3} \in \mathscr{F}, \\
\quad\left(X_{1} \backslash X_{2}\right) \cap X_{3},\left(X_{2} \backslash X_{1}\right) \cap X_{3}
\end{array}
$$

at least one of them is empty. Then the family $\mathscr{F}$ is called "perfect." Prove that if $\mathscr{F}$ is a perfect family of subsets of a given finite set $U$, then
$$
|\mathscr{F}| \leqslant |U| + 1.
$$

Answer: |\mathscr{F}|\leslant|U|+1


A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?

Answer: $\frac{15}{8}$


Let $V$ be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area $x$, where $x$ is not equal to 48 or 240. What is the sum of all possible values of $x$?

Answer: 260


Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line?

Answer: \frac{4}{3}


Compute the number of geometric sequences of length  $3$  where each number is a positive integer no larger than  $10$ .

Answer: 13


A certain product was bought in the fall, and 825 rubles were paid for it. In the fall, the price per kilogram of this product was 1 ruble cheaper than in the spring. Therefore, for the same amount in the spring, 220 kg less was bought. How much does 1 kg of the product cost in the spring, and how much was bought in the fall?

Answer: 550


Determine, with proof, the smallest positive integer  $c$  such that for any positive integer  $n$ , the decimal representation of the number  $c^n+2014$  has digits all less than  $5$ .

*Proposed by Evan Chen*

Answer: 10


In the $xOy$ coordinate plane, the curve
$$
y=(3 x-1)\left(\sqrt{9 x^{2}-6 x+5}+1\right)+(2 x-3)\left(\sqrt{4 x^{2}-12 x+13}+1\right)
$$

intersects the $x$-axis at the point $\qquad$ .

Answer: \frac{4}{5}


Three rods (with negligible thickness) are fixed perpendicularly to each other at a common point at one end. The lengths of the rods are 1, 2, and 3. This construction is placed on a table in such a way that the free ends of the rods lie on the plane of the tabletop. Determine the exact height of the fixed point above the table.

Answer: \frac{6}{7}


Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]

[i]

Answer: {f(x) = \frac{1}{x}}


The square quilt block shown is made from nine unit squares, some of which have been divided in half to form triangles. What fraction of the square quilt is shaded? Express your answer as a common fraction.

[asy]size(75);
fill(scale(3)*unitsquare,gray(.6));
path[] interior = (1,0)--(0,1)--(1,1)--cycle^^(1,1)--(2,1)--(2,2)--(1,2)--cycle^^(1,2)--(0,2)--(1,3)--cycle^^(2,1)--(2,0)--(3,1)--cycle^^(2,2)--(2,3)--(3,2)--cycle;
clip(interior);
draw(interior);
draw(scale(3)*unitsquare);
[/asy]

Answer: \frac13


A regular quadrilateral pyramid is given. The side of the base is 6, and the length of a lateral edge is 5. Sphere \( Q_{1} \) is inscribed in the pyramid. Sphere \( Q_{2} \) touches \( Q_{1} \) and all the lateral faces of the pyramid. Find the radius of sphere \( Q_{2} \).

Answer: \frac{3\sqrt{7}}{49}


Given a segment \( AB \), several other segments are placed inside it. These segments together cover the segment \( AB \). Prove that if you discard one half (left or right) of each segment, the remaining halves will cover at least \( 1/3 \) of the length of \( AB \).

Answer: \frac{1}{3}


It is given that $2^{333}$ is a 101-digit number whose first digit is 1.  How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4?

Answer: 32


Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.

Answer: 24


Let \( P M O \) be a triangle with \( P M = 2 \) and \( \angle P M O = 120^\circ \). Let \( B \) be a point on \( P O \) such that \( P M \) is perpendicular to \( M B \), and suppose that \( P M = B O \). The product of the lengths of the sides of the triangle can be expressed in the form \( a + b \sqrt[3]{c} \), where \( a, b, c \) are positive integers, and \( c \) is minimized. Find \( a + b + c \).

Answer: 28


Find all positive integers \( n \) such that
$$
\frac{n^{3n-2} - 3n + 1}{3n-2} \in \mathbf{Z}.
$$

Answer: 1


What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$?

Answer: 25


There are 10 cups on a table, 5 of them with the opening facing up and 5 with the opening facing down. Each move involves flipping 3 cups simultaneously. What is the minimum number of moves required to make all cup openings face the same direction?

Answer: 3


The sides of a non-degenerate isosceles triangle are \(x\), \(x\), and \(24\) units. How many integer values of \(x\) are possible?

Answer: 11


Find all functions from the set of positive integers to the set of positive integers such that:

$$
m^{2} + f(n) \mid m f(m) + n
$$

for all \( m, n > 0 \).

Answer: f(n)=n


Polina went to the cafeteria and saw that 2 puff pastries cost 3 times more than 2 pies. Polina didn't have enough money for 2 puff pastries, but she had enough for 1 pie and 1 puff pastry. She then wondered how many times more money she spent buying 1 puff pastry and 1 pie instead of 2 pies. Help Polina answer this question.

Answer: 2


In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles?

Answer: 30


Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$

Answer: 100


A manager schedules a consultation session at a café with two assistant managers but forgets to set a specific time. All three aim to arrive randomly between 2:00 and 5:00 p.m. The manager, upon arriving, will leave if both assistant managers are not present. Each assistant manager is prepared to wait for 1.5 hours for the other to arrive, after which they will leave if the other has not yet arrived. Determine the probability that the consultation session successfully takes place.

Answer: \frac{1}{4}


Answer: 10 + \frac{15}{2}\pi


Answer: 3


Compute
$$
\sum_{k=0}^{100}\left\lfloor\frac{2^{100}}{2^{50}+2^{k}}\right\rfloor .
$$
(Here, if \( x \) is a real number, then \( \lfloor x \rfloor \) denotes the largest integer less than or equal to \( x \).)

Answer: 101\cdot2^{49}-50


Find the minimum value of the function

$$
f(x)=(x-1)^{2}+(x-3)^{2}+\ldots+(x-101)^{2}
$$

If the result is not an integer, round it to the nearest whole number.

Answer: 44200


Let real numbers \( x_{1}, x_{2}, \ldots, x_{1997} \) satisfy the following two conditions:
1. \( -\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3} \quad(i=1, 2, \ldots, 1997) \);
2. \( x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3} \).

Find the maximum value of \( x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12} \) and explain the reasoning.

Answer: 189548


If $a$ is an element randomly selected from the set $\{1, 2, 3, 4, 5, 6, 7\}$, then the probability that the circle $C: x^2 + (y-2)^2 = 1$ is contained inside the circle $O: x^2 + y^2 = a^2$ is ______.

Answer: \frac{4}{7}


Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the perimeter of $\triangle ADE$.

Answer: 13


Given the real number \( p \geq \frac{1}{4} \), find all positive real number solutions \( x \) to the following equation:
$$
\log _{\sqrt{2}}^{2} x + 2 \log _{\sqrt{2}} x + 2 \log _{\sqrt{2}} (x^{2} + p) + p + \frac{15}{4} = 0.
$$

Answer: \frac{1}{2}


Find the largest real number  $c$  such that \[\sum_{i=1}^{101}x_i^2\geq cM^2\] whenever  $x_1,\ldots,x_{101}$  are real numbers such that  $x_1+\cdots+x_{101}=0$  and  $M$  is the median of  $x_1,\ldots,x_{101}$ .

Answer: \frac{5151}{50}


In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).

Answer: 3331


Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it is always true that \( x y z \mid (x+y+z)^{n} \).

Answer: 13


How to determine the number of fish in a pond?

We catch \( n_{a} = 20 \) fish, mark them, and release them back into the pond. After some time, enough for the marked fish to distribute themselves evenly throughout the pond, we catch \( m = 50 \) fish. Suppose among them, \( k_{1} = 7 \) are marked. Determine the number of fish in the pond \( N \) using the maximum likelihood method.

Answer: 142


Show that  $r = 2$  is the largest real number  $r$  which satisfies the following condition:

If a sequence  $a_1$ ,  $a_2$ ,  $\ldots$  of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]
for every positive integer  $n$ , then there exists a positive integer  $M$  such that  $a_{n+2} = a_n$  for every  $n \geq M$ .

Answer: 2


Given a function \( f : \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the following conditions:
1. For all \( x, y \in \mathbf{R} \), \( f(x) + f(y) + 1 \geqslant f(x + y) \geqslant f(x) + f(y) \).
2. For all \( x \in [0,1) \), \( f(0) \geqslant f(x) \).
3. \( -f(-1) = f(1) = 1 \).

Find all functions \( f \) that satisfy these conditions.

Answer: f(x)=\lfloorx\rfloor


Given a sequence $\{a_n\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \in \mathbb{N}^*$), the maximum value of $S_n - S_m$ is __.

Answer: 10


Two ferries simultaneously leave from opposite banks of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant but not equal. The ferries meet at a distance of 720 meters from one shore and continue moving. On their return trip, they meet 400 meters from the other shore. What is the width of the river?

Answer: 1760\,


Let  $ABC$  be a triangle, and  $K$  and  $L$  be points on  $AB$  such that  $\angle ACK = \angle KCL = \angle LCB$ . Let  $M$  be a point in  $BC$  such that  $\angle MKC = \angle BKM$ . If  $ML$  is the angle bisector of  $\angle KMB$ , find  $\angle MLC$ .

Answer: 30


A student over 5 years of study passed 31 exams. Each subsequent year, he took more exams than in the previous year. In the fifth year, the number of exams was three times higher than in the first year. How many exams did he have in the fourth year?

Could the student have had only one exam in his first year? Check and conclude. Could he have had two exams initially? Check and conclude. What about three exams? Perform calculations and conclude. Could the student have had 4, 5, or more exams in the first year? Why?

Answer: 8


Let  $AC$  be a diameter of a circle and  $AB$  be tangent to the circle. The segment  $BC$  intersects the circle again at  $D$ . Show that if  $AC = 1$ ,  $AB = a$ , and  $CD = b$ , then   $$ \frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2} $$

Answer: \frac{1}{^2+\frac{1}{2}}<\frac{b}{}<\frac{1}{^2}


$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\prime} B C D, A B^{\prime} C D$, $A B C^{\prime} D$, and $A B C D^{\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$?

Answer: \frac{125}{27}


Given the domain of the function $f(x)$ is $(4a-3,3-2a^{2})$, where $a\in \mathbb{R}$, and $y=f(2x-3)$ is an even function. If $B_{n}=1\times a^{1}+4\times a^{2}+7\times a^{3}+\cdots +(3n-2)a^{n}$, then $B_{50}=$ ?

Answer: 75


In $\triangle ABC$, $BD = DE = EC$ and $CF:AC = 1:3$. If the area of $\triangle ADH$ is 24 square centimeters more than the area of $\triangle HEF$, find the area of $\triangle ABC$ in square centimeters.

Answer: 108


Rachelle picks a positive integer \( a \) and writes it next to itself to obtain a new positive integer \( b \). For instance, if \( a=17 \), then \( b=1717 \). To her surprise, she finds that \( b \) is a multiple of \( a^{2} \). Find the product of all the possible values of \( \frac{b}{a^{2}} \).

Answer: 77


Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order.

Answer: 72


During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games played during the week if it is known that at least two games did not involve Vasya, Petya, or Kolya?

Answer: 30


Given the ellipse $C:\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1 \left( a > b > 0 \right)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and point $A(1,\sqrt{2})$ is on the ellipse.

$(1)$ Find the equation of ellipse $C$;

$(2)$ If a line $l$ with a slope of $\sqrt{2}$ intersects the ellipse $C$ at two distinct points $B$ and $C$, find the maximum area of $\Delta ABC$.

Answer: \sqrt{2}


Let  $S$  be area of the given parallelogram  $ABCD$  and the points  $E,F$  belong to  $BC$  and  $AD$ , respectively, such that  $BC = 3BE, 3AD = 4AF$ . Let  $O$  be the intersection of  $AE$  and  $BF$ . Each straightline of  $AE$  and  $BF$  meets that of  $CD$  at points  $M$  and  $N$ , respectively. Determine area of triangle  $MON$ .

Answer: \frac{50}{39}S


In $\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} \qquad \textbf{(E)}\ 4.5$

Answer: \sqrt{17}


For which real values of m are the simultaneous equations
\begin{align*}y &= mx + 3 \\  y& = (2m - 1)x + 4\end{align*}
satisfied by at least one pair of real numbers $(x,y)$?
$\textbf{(A)}\ \text{all }m\qquad \textbf{(B)}\ \text{all }m\neq 0\qquad \textbf{(C)}\ \text{all }m\neq 1/2\qquad \textbf{(D)}\  \text{all }m\neq 1\qquad \textbf{(E)}\ \text{no values of }m$

Answer: \textbf{(D)}\all\ne1


Given that the function $f(x) = \frac{1}{2}\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) is an even function, and points P and Q are the highest and lowest points respectively on the graph of $y = f(x)$ such that $| \overrightarrow{PQ} | = \sqrt{2}$,
(1) Find the explicit formula of the function $f(x)$;
(2) In triangle $\triangle ABC$, where $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, given that $a=1, b= \sqrt{2}$, and $f\left(\frac{A}{\pi}\right) = \frac{\sqrt{3}}{4}$, determine the size of angle $C$.

Answer: \frac{\pi}{12}


There are  $110$  guinea pigs for each of the  $110$  species, arranging as a  $110\times 110$  array. Find the maximum integer  $n$  such that, no matter how the guinea pigs align, we can always find a column or a row of  $110$  guinea pigs containing at least  $n$  different species.

Answer: 11


Prokop constructed a triangle \(ABC\), where the internal angle at vertex \(A\) was greater than \(60^{\circ}\) and the internal angle at vertex \(B\) was less than \(60^{\circ}\). Jirka drew a point \(D\) in the half-plane bounded by line \(AB\) and point \(C\), in such a way that triangle \(ABD\) was equilateral. The boys then discovered that triangles \(ACD\) and \(BCD\) are isosceles with the principal vertex at \(D\).

Determine the measure of angle \(ACB\).

Answer: 30


A snake of length \( k \) is an animal which occupies an ordered \( k \)-tuple \(\left(s_{1}, \ldots, s_{k}\right)\) of cells in an \( n \times n \) grid of square unit cells. These cells must be pairwise distinct, and \( s_{i} \) and \( s_{i+1} \) must share a side for \( i=1, \ldots, k-1 \). If the snake is currently occupying \(\left(s_{1}, \ldots, s_{k}\right)\) and \( s \) is an unoccupied cell sharing a side with \( s_{1} \), the snake can move to occupy \(\left(s, s_{1}, \ldots, s_{k-1}\right)\) instead. The snake has turned around if it occupied \(\left(s_{1}, s_{2}, \ldots, s_{k}\right)\) at the beginning, but after a finite number of moves occupies \(\left(s_{k}, s_{k-1}, \ldots, s_{1} \right)\) instead.

Determine whether there exists an integer \( n>1 \) such that one can place some snake of length at least \( 0.9 n^{2} \) in an \( n \times n \) grid which can turn around.

Answer: Yes


Rabbit and Tortoise start at point $A$ and travel to a forest amusement park at the same time. The Rabbit hops 36 meters forward in 1 minute and plays at the same spot every 3 minutes. The first break lasts 0.5 minutes, the second break lasts 1 minute, the third break lasts 1.5 minutes, and the $k$-th break lasts $0.5 k$ minutes. The Tortoise, however, never stops to rest or play along the way. It is known that the Tortoise reaches the amusement park 3 minutes and 20 seconds before the Rabbit. The distance from point $A$ to the amusement park is 2640 meters. How many meters does the Tortoise crawl in 1 minute?

Answer: 12


Three thousandths is equal to:
(A) 300
(B) 0.3
(C) 0.03
(D) 30
(E) 0.003

Answer: 0.003


The number of games won by five baseball teams are displayed on a chart. The team names are missing, and we have several clues to help identify them:

1. The Sharks won fewer games than the Raptors.
2. The Royals won more games than the Dragons, but fewer games than the Knights.
3. The Dragons won more than 30 games.

How many games did the Royals win? The teams’ wins are from a chart showing the following numbers of wins: 45, 35, 40, 50, and 60 games.

Answer: 50


Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Answer: 389


Given an integer $n \geq 2$ and a real number $a$ such that $0 < a < \frac{n+1}{n-1}$, and a complex number $z$ that satisfies the equation $z^{n+1} - a z^{n} + a z - 1 = 0$. Prove that $|z| = 1$.

Answer: |z|=1


In a competition of cheerful and resourceful participants, each correctly completed task earned 9 points, while each uncompleted or incorrectly completed task deducted 5 points. It is known that the team was given no more than 15 tasks, and they scored 57 points. How many tasks did the team complete correctly?

Answer: 8


Answer: -\frac{5}{3}


Let \( n \) be a three-digit integer with nonzero digits, not all of which are the same. Define \( f(n) \) to be the greatest common divisor of the six integers formed by any permutation of \( n \)'s digits. For example, \( f(123) = 3 \), because \(\operatorname{gcd}(123,132,213,231,312,321) = 3\). Let the maximum possible value of \( f(n) \) be \( k \). Find the sum of all \( n \) for which \( f(n) = k \).

Answer: 5994


Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at points $M$ and $N$. $P$ is a point on line $l$. When $\angle APB$ is maximized, point $P$ is exactly at $M$ (or $N$). Determine the eccentricity of the hyperbola.

Answer: \sqrt{2}


Convert the binary number $11011001_2$ to base 4.

Answer: 3121_4


Answer: 128


Two positive integers \( x \) and \( y \) are such that:

\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]

Find the smallest possible value for the sum \( x + y \).

Answer: 8044


In a triangle \( \triangle ABC \), let \( BC = a \), \( AC = b \), and \( AB = c \). If \( \angle A + \angle C = 2 \angle B \), find the greatest positive integer \( n \) such that \( a^n + c^n \leq 2b^n \) holds for any such triangle.

Answer: 4


Let  $G=G(V,E)$  be a simple graph with vertex set  $V$  and edge set  $E$ . Suppose  $|V|=n$ . A map  $f:\,V\rightarrow\mathbb{Z}$  is called good, if  $f$  satisfies the followings:
(1)  $\sum_{v\in V} f(v)=|E|$ ;
(2) color arbitarily some vertices into red, one can always find a red vertex  $v$  such that  $f(v)$  is no more than the number of uncolored vertices adjacent to  $v$ .
Let  $m(G)$  be the number of good maps. Prove that if every vertex in  $G$  is adjacent to at least one another vertex, then  $n\leq m(G)\leq n!$ .

Answer: n\le(G)\len!


The sum of nine different natural numbers is 200. Is it always possible to select four numbers from them such that their sum is greater than 100?

Answer: No


Let  $P$  and  $Q$  be the midpoints of sides  $AB$  and  $BC,$  respectively, of  $\triangle ABC.$  Suppose  $\angle A = 30^{\circ}$  and  $\angle PQC = 110^{\circ}.$  Find  $\angle B$  in degrees.

Answer: 80


Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\prod_{i=1}^6 h(y_i)$.

Answer: 676


Suppose  $m$  and  $n$  are relatively prime positive integers. A regular  $m$ -gon and a regular $n$ -gon are inscribed in a circle. Let  $d$  be the minimum distance in degrees (of the arc along
the circle) between a vertex of the  $m$ -gon and a vertex of the  $n$ -gon. What is the maximum
possible value of  $d$ ?

Answer: \frac{180}{mn}


Suppose  $a_{1} < a_{2}< \cdots < a_{2024}$  is an arithmetic sequence of positive integers, and   $b_{1} <b_{2} < \cdots <b_{2024}$  is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

*Ray Li*

Answer: 11


Answer: \frac{243}{32}


In $\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is
[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B); draw(A--P--C--A--B--C); label("A", A, W); label("B", B, E); label("C", C, NE); label("P", P, NW); label("6", 3*dir(A--C), SE); label("7", B+3*dir(B--C), NE); label("8", (4,0), S); [/asy]
$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 11$

Answer: 9


The base of a pyramid is a rhombus, one of whose angles is \(\alpha\). The lateral faces are equally inclined to the base. Through the midpoints of two adjacent sides of the base and the pyramid's apex, a plane is passed, which forms an angle \(\beta\) with the base. The area of the resultant cross-section is \(S\). Find the side length of the rhombus.

Answer: 2\sqrt{\frac{2S\cos\beta}{\sin\alpha}}


Answer: 10


A plane passing through the midpoints of edges $AB$ and $CD$ of the triangular pyramid $ABCD$ divides edge $AD$ in the ratio 3:1, measured from vertex $A$. In what ratio does this plane divide edge $BC$?

Answer: 3:1


Given a point P on the parabola $y^2=4x$, let the distance from point P to the directrix of this parabola be $d_1$, and the distance from point P to the line $x+2y-12=0$ be $d_2$.

Answer: \frac{11 \sqrt{5}}{5}


A triangular wire frame with side lengths of $13, 14, 15$ is fitted over a sphere with a radius of 10. Find the distance between the plane containing the triangle and the center of the sphere.

Answer: 2\sqrt{21}


In the number \( A \), the digits are in ascending order (from left to right). What is the sum of the digits of the number \( 9 \cdot A \)?

Answer: 9


There are a total of 100 parts in 7 boxes. Any two of these boxes contain a different number of parts. Is it necessary to find among them 3 boxes that contain at least 50 parts?

Answer: Yes


Answer: \frac{\pi}{2}


A motorcyclist traveled from point $A$ to point $B$, which is 120 km away from $A$. On the return trip, he drove at the same initial speed but had to stop for 10 minutes after driving for one hour. After the stop, he continued to point $A$ at a speed that was 6 km/h faster than his initial speed. What was the motorcyclist's initial speed, given that the return trip took the same amount of time as the trip from $A$ to $B$?

Answer: 48\,


There are people with masses $50, 51, 55, 57, 58, 59, 60, 63, 75,$ and $140$ kg standing in front of an elevator. The elevator has a load capacity of $180$ kg. What is the minimum number of trips necessary to ensure that all people can be transported?

Answer: 4


In a certain month, three Saturdays fell on even dates. What day of the week was the 25th of the month?

Answer: Monday


In a $2009 \times 2009$ grid, each small square is filled with a real number whose absolute value does not exceed 1. In addition, for every $2 \times 2$ square, the sum of the 4 real numbers is 0. Find the maximum sum of all the numbers in the grid.

Answer: 2009


Xiao Ming participated in the "Inheriting Classics, Building the Future" themed speech competition. His scores for speech image, speech content, and speech effect were 9, 8, and 9 respectively. If the scores for speech image, speech content, and speech effect are determined in a ratio of 2:5:3 to calculate the final score, then Xiao Ming's final competition score is ______ points.

Answer: 8.5


Find the smallest three-digit number with the property that if you append to it a number that is greater by 1 on the right, the result (a six-digit number) will be a perfect square.

Answer: 183


A point  $X$  exactly  $\sqrt{2}-\frac{\sqrt{6}}{3}$  away from the origin is chosen randomly. A point  $Y$  less than  $4$  away from the origin is chosen randomly. The probability that a point  $Z$  less than  $2$  away from the origin exists such that  $\triangle XYZ$  is an equilateral triangle can be expressed as  $\frac{a\pi + b}{c \pi}$  for some positive integers  $a, b, c$  with  $a$  and  $c$  relatively prime. Find  $a+b+c$ .

Answer: 34


Find the number of solutions to the equation:  $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x.

Answer: 5


I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?

Answer: 1


Given that the real numbers \( x \) and \( y \) satisfy the inequality \(\left(x+\sqrt{x^{2}+1}\right)\left(y+\sqrt{y^{2}+1}\right) \geqslant 1\), find the minimum value of \( x + y \).

Answer: 0


Find all positive integers  $n$  for which there exist real numbers  $x_1, x_2,. . . , x_n$  satisfying all of the following conditions:
(i)  $-1 <x_i <1,$  for all  $1\leq  i \leq  n.$ (ii)  $ x_1 + x_2 + ... + x_n = 0.$ (iii)  $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

Answer: 1


Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

Answer: \frac{37}{56}


Given that line \( l \) passes through the left focus \( F \) of the ellipse \( C: \frac{x^{2}}{2} + y^{2} = 1 \) and intersects the ellipse \( C \) at points \( A \) and \( B \). Let \( O \) be the origin. If \( O A \) is perpendicular to \( O B \), find the distance from the point \( O \) to the line \( A B \).

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


Let \( z = e^{\frac{2 \pi i}{101}} \) and \( \omega = e^{\frac{2 \pi i}{10}} \). Prove that
$$
\prod_{a=0}^{9} \prod_{b=0}^{100} \prod_{c=0}^{100}\left(\omega^{a} + z^{b} + z^{c}\right)
$$
is an integer and find (with proof) its remainder upon division by 101.

Answer: 13


Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at  $0$ . Every minute, he has a  $\frac{1}{3}$  chance of moving  $1$  unit left, a  $\frac{1}{3}$  chance of moving  $1$  unit right, and  $\frac{1}{3}$  chance of getting eaten. Find the expected number of times Kelvin returns to  $0$  (not including the start) before he gets eaten.

Answer: \frac{3\sqrt{5}-5}{5}


In a recent Cych Cup math competition, the maximum score was 100 points. Among the participating students, the highest score was 83 points, and the lowest score was 30 points (all scores are integers). There were 8,000 students in total participating in the competition. How many students at least have the same score?

Answer: 149


Suppose real numbers $\lambda_{i}$ satisfy $0<\lambda_{i}<1$ for $i=1, 2, 3$. Define $X=\lambda_{1}\left(1-\lambda_{3}\right)$, $Y=\lambda_{2}\left(1-\lambda_{1}\right)$, and $Z=\lambda_{3}\left(1-\lambda_{2}\right)$. Prove that at least one of $X, Y, Z$ must be less than or equal to $1-(X+Y+Z)$. (Hint: To solve this algebra problem, try constructing a geometric model by approaching it from the opposite direction.)

Answer: 1


Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored?

Answer: 150


Let  $ n > 1$  be an odd positive integer and  $ A = (a_{ij})_{i, j = 1..n}$  be the  $ n \times n$  matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j  1 & \text{if }i-j \equiv \pm 2 \pmod n  0 & \text{otherwise}\end{cases}.\]
Find  $ \det A$ .

Answer: 4


Suppose a parabola has vertex $\left(\frac{3}{2},-\frac{25}{4}\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$.

Answer: -2


If point $P$ is the golden section point of segment $AB$, and $AP < BP$, $BP=10$, then $AP=\_\_\_\_\_\_$.

Answer: 5\sqrt{5} - 5


A tetrahedron ABCD has edges of total length 1. The angles at vertex A (angles BAC, BAD, and CAD) are all 90 degrees. Find the maximum volume of the tetrahedron.

Answer: \frac{5\sqrt{2}-7}{162}


A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$.

Answer: 2.7


There exist positive integers $a,$ $b,$ and $c$ such that
\[3 \sqrt{\sqrt[3]{5} - \sqrt[3]{4}} = \sqrt[3]{a} + \sqrt[3]{b} - \sqrt[3]{c}.\]Find $a + b + c.$

Answer: 47


Given that the digits of the integers must be from the set $\{1,3,4,5,6,8\}$ and the integers must be between $300$ and $800$, determine how many even integers have all distinct digits.

Answer: 40


23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85.
Ian sighs, "I am even older than that number."
Determine Ian's age.

Answer: 14


Let \( x_{1}, \ldots, x_{100} \) be positive real numbers such that \( x_{i} + x_{i+1} + x_{i+2} \leq 1 \) for all \( i \leq 100 \) (with the convention that \( x_{101} = x_{1} \) and \( x_{102} = x_{2} \)). Determine the maximum value of

\[
S = \sum_{i=1}^{100} x_{i} x_{i+2}
\]

Answer: \frac{25}{2}


Given the equation csa\sin (+ \dfrac {\pi}{6})+siasn(a- \dfrac {\pi}{3})=, evaluate the expression.

Answer: \dfrac{1}{2}


In the cells of an $80 \times 80$ table, pairwise distinct natural numbers are placed. Each number is either prime or the product of two prime numbers (possibly the same). It is known that for any number $a$ in the table, there is a number $b$ in the same row or column such that $a$ and $b$ are not coprime. What is the largest possible number of prime numbers that can be in the table?

Answer: 4266


Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\).

Answer: 180


Let $a, b, c$ be positive real numbers such that $a+b+c=10$ and $a b+b c+c a=25$. Let $m=\min \{a b, b c, c a\}$. Find the largest possible value of $m$.

Answer: \frac{25}{9}


Find the smallest positive integer \( k \) such that \( (k-10)^{5026} \geq 2013^{2013} \).

Answer: 55


A lattice point in the plane is a point of the form \((n, m)\), where \(n\) and \(m\) are integers. Consider a set \(S\) of lattice points. We construct the transform of \(S\), denoted by \(S^{\prime}\), by the following rule: the pair \((n, m)\) is in \(S^{\prime}\) if and only if any of \((n, m-1)\), \((n, m+1)\), \((n-1, m)\), \((n+1, m)\), or \((n, m)\) is in \(S\). How many elements are in the set obtained by successively transforming \(\{(0,0)\}\) 14 times?

Answer: 421


While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

Answer: 90


(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this
angle (There was an error in published condition of this problem).

Answer: 120


Given the geometric sequence $\{a_n\}$, $a_5a_7=2$, $a_2+a_{10}=3$, determine the value of $\frac{a_{12}}{a_4}$.

Answer: \frac {1}{2}


Xiao Ming collected 20 pieces of data in a survey, as follows:
$95\ \ \ 91\ \ \ 93\ \ \ 95\ \ \ 97\ \ \ 99\ \ \ 95\ \ \ 98\ \ \ 90\ \ \ 99$
$96\ \ \ 94\ \ \ 95\ \ \ 97\ \ \ 96\ \ \ 92\ \ \ 94\ \ \ 95\ \ \ 96\ \ \ 98$
$(1)$ When constructing a frequency distribution table with a class interval of $2$, how many classes should it be divided into?
$(2)$ What is the frequency and relative frequency of the class interval $94.5\sim 96.5$?

Answer: 0.4


A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.

Answer: \frac{60}{11}


Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form.

Answer: 12


Let $ m\equal{}\left(abab\right)$ and $ n\equal{}\left(cdcd\right)$ be four-digit numbers in decimal system. If $ m\plus{}n$ is a perfect square, find the largest value of $ a\cdot b\cdot c\cdot d$.

Answer: 600


Let  $n$  and  $k$  be positive integers. Chef Kao cuts a circular pizza through  $k$  diameters, dividing the pizza into  $2k$  equal pieces. Then, he dresses the pizza with  $n$  toppings. For each topping, he chooses  $k$  consecutive pieces of pizza and puts that topping on all of the chosen pieces. Then, for each piece of pizza, Chef Kao counts the number of distinct toppings on it, yielding  $2k$  numbers. Among these numbers, let  $m$  and  $M$  being the minimum and maximum, respectively. Prove that  $m + M = n$ .

Answer: +M=n


In an arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$, and $a_{11} > |a_{10}|$. The maximum negative value of the partial sum $S_n$ of the first $n$ terms of the sequence $\{a_n\}$ is the sum of the first ______ terms.

Answer: 19


Using the equality \( \lg 11 = 1.0413 \), find the smallest number \( n > 1 \) for which among the \( n \)-digit numbers, there is not a single one equal to some natural power of the number 11.

Answer: 26


In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways can this reassignment be made?

Answer: 0


On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: "both my neighbors are liars," while the remaining said: "among my neighbors, there is exactly one liar." What is the maximum number of knights that can sit at this table?

Answer: 67


Given four points \(O, A, B, C\) on a plane, such that \(OA = 4\), \(OB = 3\), \(OC = 2\), and \(\overrightarrow{OB} \cdot \overrightarrow{OC} = 3\), find the maximum value of the area \(S_{\triangle ABC}\).

Answer: 2\sqrt{7} + \frac{3\sqrt{3}}{2}


Let the sequence  $\{a_i\}^\infty_{i=0}$  be defined by  $a_0 =\frac12$  and  $a_n = 1 + (a_{n-1} - 1)^2$ . Find the product   $$ \prod_{i=0}^\infty a_i=a_0a_1a_2\ldots $$

Answer: \frac{2}{3}


In the plane rectangular coordinate system $xOy$, the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ passes through the points $A\left(-3,m\right)$ and $B\left(-1,n\right)$.
$(1)$ When $m=n$, find the length of the line segment $AB$ and the value of $h$;
$(2)$ If the point $C\left(1,0\right)$ also lies on the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$, and $m \lt 0 \lt n$,
① find the abscissa of the other intersection point of the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ with the $x$-axis (expressed in terms of $h$) and the range of values for $h$;
② if $a=-1$, find the area of $\triangle ABC$;
③ a line passing through point $D(0$,$h^{2})$ perpendicular to the $y$-axis intersects the parabola at points $P(x_{1}$,$y_{1})$ and $(x_{2}$,$y_{2})$ (where $P$ and $Q$ are not coincident), and intersects the line $BC$ at point $(x_{3}$,$y_{3})$. Is there a value of $a$ such that $x_{1}+x_{2}-x_{3}$ is always a constant? If so, find the value of $a$; if not, explain why.

Answer: -\frac{1}{4}


In $\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An excircle of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$, and the other two excircles are both externally tangent to $\omega$. Find the minimum possible value of the perimeter of $\triangle ABC$.

Answer: 20


By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)

Answer: 8


Given that the random variable $X$ follows a normal distribution $N(0,\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \leqslant X \leqslant 2)$.

Answer: 0.954


Let \( P \) and \( Q \) be polynomials in \( \mathbb{R}[X] \). Show that if there exists a real number \( N \) such that \( P(x) < Q(x) \) for all \( x > N \), then \(\operatorname{deg}(P) \leq \operatorname{deg}(Q)\).

Answer: \operatorname{}(P)\le\operatorname{}(Q)


Given that $P\left(x_{0}, y_{0}\right)$ is an arbitrary point on the ellipse $\Gamma$: $\frac{x^{2}}{8}+\frac{y^{2}}{2}=1$, the line $y=kx+m$ $(k m \neq 0)$ cuts a chord $MN$ in the ellipse $\Gamma_{1}: \frac{x^{2}}{4}+y^{2}=1$ such that $MN$ is bisected by the line $OP$. Given that $m y_{0} > -1$ and the area of triangle $PMN$ is 1, determine the shape of the quadrilateral $OMPN$ and the number of common points of the line $PM$ with the ellipse $\Gamma_{1}$. Prove your conclusions.

Answer: 1


Given the set \( A = \{ a \mid -1 \leqslant a \leqslant 2 \} \), find the area of the plane region \( B = \{ (x, y) \mid x, y \in A, x + y \geqslant 0 \} \).

Answer: 7


Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the volume of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Answer: 365


Given a quadrilateral \( ABCD \) with an area of 1, points \( H \) and \( G \) divide \( AD \) into \( n \) equal segments, and points \( E \) and \( F \) similarly divide \( BC \) into \( n \) equal segments. Prove that the area of quadrilateral \( EFGH \) is \(\frac{n-2}{n}\).

Answer: \frac{n-2}{n}


Find the value of
$$
\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of \(\frac{1}{2^{a} 3^{b} 5^{c}}\) over all triples of positive integers \((a, b, c)\) satisfying \(a < b < c\)).

Answer: \frac{1}{1624}


Find the smallest integer \( n > 1 \) such that \(\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}\) is a square.

Answer: 337


Through the right focus of the hyperbola \( x^{2}-\frac{y^{2}}{2}=1 \), a line \( l \) intersects the hyperbola at points \( A \) and \( B \). If the real number \( \lambda \) makes \( |AB|=\lambda \) and there are exactly 3 such lines \( l \), then \( \lambda = \) ____.

Answer: 4


A convex 2011-sided polygon is drawn on the board. Peter sequentially draws diagonals inside the polygon such that each newly drawn diagonal intersects at most one of the previously drawn diagonals at internal points. What is the maximum number of diagonals Peter can draw?

Answer: 4016


Let  $a$  and  $b$  be positive integers such that  $\text{gcd}(a, b) + \text{lcm}(a, b)$  is a multiple of  $a+1$ . If  $b \le a$ , show that  $b$  is a perfect square.

Answer: b


Answer: \frac{1}{4}


Prove that a nine-digit number that uses all digits except zero and ends in the digit 5 cannot be a perfect square of an integer.

Answer: 9


We are given a triangle  $ABC$ . Points  $D$  and  $E$  on the line  $AB$  are such that  $AD=AC$  and  $BE=BC$ , with the arrangment of points  $D - A - B - E$ . The circumscribed circles of the triangles  $DBC$  and  $EAC$  meet again at the point  $X\neq C$ , and the circumscribed circles of the triangles  $DEC$  and  $ABC$  meet again at the point  $Y\neq C$ . Find the measure of  $\angle ACB$  given the condition  $DY+EY=2XY$ .

Answer: 60


For given $n$ distinct numbers $a_{1}, a_{2}, \cdots, a_{n} \in \mathbf{N}, n \geqslant 2$, define $P_{i}=\prod_{\substack{1 \leq j \leq n \leq n \\ j \neq i}}\left(a_{i}-a_{j}\right), i=1,2, \cdots, n$. Prove that for any $k \in \mathbf{N}, \sum_{i=1}^{n} \frac{a_{i}^{k}}{P_{i}}$ is an integer.

Answer: \sum_{i=1}^{n}\frac{a_{i}^{k}}{P_{i}}


For a natural number  $b$ , let  $N(b)$  denote the number of natural numbers  $a$  for which the equation  $x^2 + ax + b = 0$  has integer roots. What is the smallest value of  $b$  for which  $N(b) = 20$ ?

Answer: 240


An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of flavors?

Answer: 92


Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

Answer: \frac{1}{5}


The length of a rectangle was increased by $12\%$, and the width was increased by $15\%$. a) By what percentage is the area of the new rectangle greater than the area of the original rectangle? b) Find the ratio of the length to the width of the original rectangle, given that the perimeter of the new rectangle is $13\%$ greater than the perimeter of the original.

Answer: 2:1


In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many scenarios are there where both A and B grab a red envelope? (Answer with a number)

Answer: 36


In triangle \(ABC\), a point \(D\) is marked on side \(AC\) such that \(BC = CD\). Find \(AD\) if it is known that \(BD = 13\) and angle \(CAB\) is three times smaller than angle \(CBA\).

Answer: 13


Joe has a collection of $23$ coins, consisting of $5$-cent coins, $10$-cent coins, and $25$-cent coins. He has $3$ more $10$-cent coins than $5$-cent coins, and the total value of his collection is $320$ cents. How many more $25$-cent coins does Joe have than $5$-cent coins?
$\textbf{(A) }   0   \qquad        \textbf{(B) }   1   \qquad    \textbf{(C) }   2   \qquad   \textbf{(D) }  3  \qquad  \textbf{(E) }   4$

Answer: \textbf{(C)}2


Determine the sum of the digits in the cube of a number composed of $n$ nines.

Answer: 18n


Given that a shop advertises everything as "half price in today's sale," and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both items that the customer ultimately pays when the second item's original price is the same as the first's.

Answer: 20\%


Dad gives Xiaojun the same amount of pocket money every day. Xiaojun originally had some amount of money. If he spends 10 yuan every day, he can last for 6 days; if he spends 15 yuan every day, he can last for 3 days. How much money did Xiaojun originally have?

Answer: 30


Find the length of the bisector of the right angle in a right triangle with legs of lengths \(a\) and \(b\).

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


Let the natural number \( n \) have the following property: If you choose 50 different numbers from the natural numbers \( 1, 2, \cdots, n \), there must be two numbers whose difference is equal to 7. Find the maximum value of \( n \).

Answer: 98


Let  $n\geq 2$  be an integer and let  $f$  be a  $4n$ -variable polynomial with real coefficients. Assume that, for any  $2n$  points  $(x_1,y_1),\dots,(x_{2n},y_{2n})$  in the Cartesian plane,  $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$  if and only if the points form the vertices of a regular  $2n$ -gon in some order, or are all equal.

Determine the smallest possible degree of  $f$ .

(Note, for example, that the degree of the polynomial   $$ g(x,y)=4x^3y^4+yx+x-2 $$   is  $7$  because  $7=3+4$ .)

*Ankan Bhattacharya*

Answer: 2n


Define a new operation: \( A * B = \frac{(A - B)}{3} \) and \( A \square B = (A + B) \times 3 \). Calculate: \( (39 * 12) \square 3 = \) .

Answer: 36


A square board is dissected into  $n^2$  rectangular cells by  $n-1$  horizontal and  $n-1$  vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of  $n$  black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.

Answer: Thetotalareaofallblackcellsisnotthanthetotalareaofallwhitecells.


A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts?

Answer: 60


Let  $PRUS$  be a trapezium such that  $\angle PSR = 2\angle QSU$  and  $\angle SPU = 2 \angle UPR$ . Let  $Q$  and  $T$  be on  $PR$  and  $SU$  respectively such that  $SQ$  and  $PU$  bisect  $\angle PSR$  and  $\angle SPU$  respectively. Let  $PT$  meet  $SQ$  at  $E$ . The line through  $E$  parallel to  $SR$  meets  $PU$  in  $F$  and the line through  $E$  parallel to  $PU$  meets  $SR$  in  $G$ . Let  $FG$  meet  $PR$  and  $SU$  in  $K$  and  $L$  respectively. Prove that  $KF$  =  $FG$  =  $GL$ .

Answer: KF=FG=GL


There are 52 students in a class. Now, using the systematic sampling method, a sample of size 4 is drawn. It is known that the seat numbers in the sample are 6, X, 30, and 42. What should be the seat number X?

Answer: 18


Let  $a,b,c$  be positive reals satisfying  $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$ .  Prove that  $a^a b^b c^c \ge 1$ .

*Proposed by Evan Chen*

Answer: ^^^\ge1


Let $n$ be the 200th smallest positive real solution to the equation $x-\frac{\pi}{2}=\tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Answer: 314


In the triangular pyramid $A B C D$ with a base $A B C$, the lateral edges are pairwise perpendicular, $D A=D B=5$, and $D C=1$. From a point on the base, a light ray is emitted. After reflecting exactly once from each of the lateral faces (without reflecting from the edges), the ray hits a point on the base of the pyramid. What is the minimum distance the ray could have traveled?

Answer: \frac{10\sqrt{3}}{9}


Given a regular triangular pyramid \(P-ABC\), where points \(P\), \(A\), \(B\), and \(C\) all lie on the surface of a sphere with radius \(\sqrt{3}\), and \(PA\), \(PB\), and \(PC\) are mutually perpendicular, find the distance from the center of the sphere to the cross-section \(ABC\).

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


Given two sides \(a\) and \(b\) of a triangle and the angle bisector \(l\) between them. Find the angle.

Answer: 2\arccos(\frac{(b)}{2ab})


Let complex numbers \( z_1 \) and \( z_2 \) satisfy \( \operatorname{Re}(z_1) > 0 \) and \( \operatorname{Re}(z_2) > 0 \), and \( \operatorname{Re}(z_1^2) = \operatorname{Re}(z_2^2) = 2 \) (where \( \operatorname{Re}(z) \) denotes the real part of the complex number \( z \)).

1. Find the minimum value of \( \operatorname{Re}(z_1 z_2) \).
2. Find the minimum value of \( |z_1 + 2| + |\overline{z_2} + 2| - |\overline{z_1} - z_2| \).

Answer: 4\sqrt{2}


Given a real number $a$ satisfying ${a}^{\frac{1}{2}}\leqslant 3$ and $\log _{a}3\leqslant \frac{1}{2}$.
$(1)$ Find the range of real number $a$;
$(2)$ If $a \gt 1$, $f\left(x\right)=mx^{a}+\ln \left(1+x\right)^{a}-a\ln \left(1-x\right)-2\left(m\in R\right)$, and $f(\frac{1}{2})=a$, find the value of $f(-\frac{1}{2})$.

Answer: -13


Find the logarithmic residue of the function

$$
f(z)=\frac{\operatorname{ch} z}{e^{i z}-1}
$$

with respect to the contour $C:|z|=8$.

Answer: 3


Determine the smallest positive real number  $r$  such that there exist differentiable functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  and  $g: \mathbb{R} \rightarrow \mathbb{R}$  satisfying
(a)  $f(0)>0$ ,
(b)  $g(0)=0$ ,
(c)  $\left|f^{\prime}(x)\right| \leq|g(x)|$  for all  $x$ ,
(d)  $\left|g^{\prime}(x)\right| \leq|f(x)|$  for all  $x$ , and
(e)  $f(r)=0$ .

Answer: \frac{\pi}{2}


Count the number of functions $f: \mathbb{Z} \rightarrow\{$ 'green', 'blue' $\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'.

Answer: 39601


From the base of a height of a regular triangular pyramid, a perpendicular equal to \( p \) is drawn to the lateral edge. Find the volume of the pyramid if the dihedral angle between its lateral faces is \( \alpha \).

Answer: \frac{9p^3\tan^3(\frac{\alpha}{2})}{4\sqrt{3\tan^2(\frac{\alpha}{2})-1}}


Given a finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers (with \(n \geq 3\)), and the recurrence relation \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\) holds for all \(\kappa \leq n-2\). The sequence must contain \(a_{k} = 2022\). Determine the maximum number of three-digit numbers that are multiples of 4 that this sequence can contain.

Answer: 225


In the arithmetic sequence $\{a\_n\}$, $a_{66} < 0$, $a_{67} > 0$, and $a_{67} > |a_{66}|$. $S_{n}$ denotes the sum of the first $n$ terms of the sequence. Find the smallest value of $n$ such that $S_{n} > 0$.

Answer: 132


The geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1}=1536$, and common ratio $q=-\frac{1}{2}$. Let $\Pi_{n}$ denote the product of its first $n$ terms $\left(n \in \mathbf{N}^{*}\right)$. Find the value of $n$ that maximizes $\Pi_{n}\$.

Answer: 11


Let $x_1$, $x_2$, ..., $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is.

Answer: 61


Jeff will pick a card at random from ten cards numbered 1 through 10.  The number on this card will indicate his starting point on the number line shown below.  He will then spin the fair spinner shown below (which has three congruent sectors) and follow the instruction indicated by his spin.  From this new point he will spin the spinner again and follow the resulting instruction.  What is the probability that he ends up at a multiple of 3 on the number line?  Express your answer as a common fraction. [asy]
import graph;
size(10cm);
defaultpen(linewidth(0.7)+fontsize(8));

xaxis(-2,13,Ticks(OmitFormat(-1),1.0,begin=false,end=false,beginlabel=false,endlabel=false),Arrows(4));

label("-1",(-1,-0.98));

real r=3.5;
pair center=(17,0);
draw(circle(center,r));
int i;
for(i=1;i<=3;++i)

{

draw(center--center+r*dir(120*i-30));

}
label("$\parbox{1cm}{move \\ 1 space \\ left}$",center+r/2*dir(150));
label("$\parbox{1cm}{move \\ 1 space \\ right}$",center+r/2*dir(270));
label("$\parbox{1cm}{move \\ 1 space \\ right}$",center+r/2*dir(30));
draw(center--center+3*r/4*dir(80),EndArrow(4));[/asy]

Answer: \frac{31}{90}


Given the line $l: x-y+4=0$ and the circle $C: \begin{cases}x=1+2\cos \theta \\ y=1+2\sin \theta\end{cases} (\theta$ is a parameter), find the distance from each point on $C$ to $l$.

Answer: 2 \sqrt{2}-2


Let $n$ be a positive integer. Determine the function $f(n)$ if $f(1)=1$, $f(2)=3$, and

$$
2 f(n-2) + f(n-1) = f(n)
$$

Answer: f(n)=\frac{2^{n+1}+(-1)^n}{3}


In the acute triangle \( \triangle ABC \), the angle bisector of angle \( A \) intersects the circumcircle of the triangle at another point \( A_1 \); similarly define points \( B_1 \) and \( C_1 \). The line \( AA_1 \) intersects the external angle bisectors of angles \( B \) and \( C \) at point \( A_0 \); similarly define points \( B_0 \) and \( C_0 \). Prove that:
1. The area of \( \triangle A_0B_0C_0 \) is twice the area of the hexagon \( AC_1BA_1CB_1 \).
2. The area of \( \triangle A_0B_0C_0 \) is at least four times the area of \( \triangle ABC \).
(IMO - 30th problem)

Answer: 4


The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds.

Answer: 12.8


Find all \( x \) in the interval \([0, 2\pi]\) which satisfy:
\[ 2 \cos x \leq \left| \sqrt{1 + \sin 2x} - \sqrt{1 - \sin 2x} \right| \leq \sqrt{2}. \]

Answer: [\frac{\pi}{4},\frac{7\pi}{4}]


How many ways are there to insert  $+$ 's between the digits of  $111111111111111$  (fifteen  $1$ 's) so that the result will be a multiple of  $30$ ?

Answer: 2002


The Nanjing Youth Olympic Games are about to open, and a clothing store owner, Mr. Chen, spent 3600 yuan to purchase two types of sportswear, A and B, and sold them out quickly. When Mr. Chen went to purchase the same types and quantities of clothing again, he found that the purchase prices of types A and B had increased by 20 yuan/piece and 5 yuan/piece, respectively, resulting in an additional expenditure of 400 yuan. Let the number of type A clothing purchased by Mr. Chen each time be $x$ pieces, and the number of type B clothing be $y$ pieces.
(1) Please write down the function relationship between $y$ and $x$ directly: .
(2) After calculating, Mr. Chen found that the average unit price of types A and B clothing had increased by 8 yuan during the second purchase compared to the first.
  ① Find the values of $x$ and $y$.
  ② After selling all the clothing purchased for the second time at a 35% profit, Mr. Chen took all the sales proceeds to purchase more goods. At this time, the prices of both types of clothing had returned to their original prices, so Mr. Chen spent 3000 yuan to purchase type B clothing, and the rest of the money was used to purchase type A clothing. As a result, the quantities of types A and B clothing purchased were exactly equal. How many pieces of clothing did Mr. Chen purchase in total this time?

Answer: 80


Find the area of triangle $ABC$ given below:

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy]

Answer: \frac{1}{4}


For which values of \( x \) is the expression \( 4^{\sin ^{2} x} + 8^{\cos ^{2} x} \) minimized? What is the minimum value?

Answer: 5\cdot\sqrt[5]{\frac{16}{27}}


$A$  and  $B$  are on a circle of radius  $20$  centered at  $C$ , and  $\angle ACB = 60^\circ$ .  $D$  is chosen so that  $D$  is also on the circle,  $\angle ACD = 160^\circ$ , and  $\angle DCB = 100^\circ$ . Let  $E$  be the intersection of lines  $AC$  and  $BD$ . What is  $DE$ ?

Answer: 20


There are 10 students standing in a row. A teacher wants to assign a hat to each student, with the hats available in two colors: red and white. There are more than 10 hats of each color available. The requirement is that in any consecutive group of students, the difference between the number of students wearing red hats and the number of students wearing white hats is at most 2. How many ways can the teacher assign the hats?

Answer: 94


Given point \( A \) is a point inside a unit circle centered at \( O \), satisfying \( |\overrightarrow{OA}| = \frac{1}{2} \). Points \( B \) and \( C \) are any two points on the unit circle \( O \). Determine the range of values for \( \overrightarrow{AC} \cdot \overrightarrow{BC} \).

Answer: [-\frac{1}{8},3]


Consider positive integers $a \leq b \leq c \leq d \leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of $N$?

Answer: 28


Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is

Answer: 36


A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in the display.

Answer: 24


What is the smallest natural number such that removing its first digit results in a number that is four times a prime number, and the number following the result is five times a prime number?

Answer: 1964


Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
$\textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\ \qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\ \qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$

Answer: \textbf{(E)}\


Determine all real values of the parameter \(a\) for which the equation
\[ 16x^{4} - ax^{3} + (2a + 17)x^{2} - ax + 16 = 0 \]
has exactly four distinct real roots that form a geometric progression.

Answer: 170


Given that \(a, b, c, d\) are positive integers, \(r = 1 - \frac{a}{b} - \frac{c}{d}\), and if \(a + c \leq 1993\) and \(r > 0\), prove that \(r > \frac{1}{1993^3}\).

Answer: r>\frac{1}{1993^3}


In the Cartesian coordinate system $xOy$, curve $C_{1}: y^{2}=4x$ and curve $C_{2}:(x-4)^{2}+y^{2}=8$. A line $l$ with a slope of $45^\circ$ passes through a point $P$ on $C_{1}$ and intersects $C_{2}$ at two distinct points $Q$ and $R$. Find the range of values for $|PQ| \cdot |PR|$.

Answer: [4,8)\cup(8,200)


Suppose \(ABC\) is a triangle with circumcenter \(O\) and orthocenter \(H\) such that \(A\), \(B\), \(C\), \(O\), and \(H\) are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of \(ABC\)?

Answer: \sqrt{10}


The sum of five consecutive integers is 2015. What is the smallest of these integers?
A 401
B 403
C 405
D 407
E 409

Answer: 401


If each of the three operation signs, $+$, $\text{--}$, $\times$, is used exactly ONCE in one of the blanks in the expression
\[5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3\]
then the value of the result could equal
$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 19$

Answer: (E)\19


In a \( 29 \times 29 \) table, the numbers \( 1, 2, 3, \ldots, 29 \) are each written 29 times. It is found that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.

Answer: 15


Two athletes start running simultaneously - the first from \( A \) to \( B \), and the second from \( B \) to \( A \). They run at different but constant speeds and meet at a distance of 300 meters from \( A \). After running the track \( AB \) to the end, each of them immediately turns back and meets the other again at a distance of 400 meters from \( B \). Find the length of \( AB \).

Answer: 500


Given triangle  $ABC$  and a point  $P$  inside it,  $\angle BAP=18^\circ$ ,  $\angle CAP=30^\circ$ ,  $\angle ACP=48^\circ$ , and  $AP=BC$ .  If  $\angle BCP=x^\circ$ , find  $x$ .

Answer: 6


Let  $S$  be the sum of all positive integers that can be expressed in the form  $2^a \cdot 3^b \cdot 5^c$ , where  $a$ ,  $b$ ,  $c$  are positive integers that satisfy  $a+b+c=10$ . Find the remainder when  $S$  is divided by  $1001$ .

*Proposed by Michael Ren*

Answer: 34


Each of  $n$  boys and  $n$  girls chooses a random number from the set  $\{ 1, 2, 3, 4, 5 \}$ , uniformly and independently.  Let  $p_n$  be the probability that every boy chooses a different number than every girl.  As  $n$  approaches infinity, what value does  $\sqrt[n]{p_n}$  approach?

Answer: \frac{6}{25}


Prove that number  $1$  can be represented as a sum of a finite number  $n$  of real numbers, less than  $1,$  not necessarily  distinct, which contain in their decimal representation only the digits  $0$  and/or  $7.$  Which is the least possible number  $n$ ?

Answer: 8


Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$.

Answer: -\frac{\pi}{8}


Answer: \frac{140}{\sqrt{1021}}


Given the perpendicular lines $e_{1}$ and $e_{2}$, and a point $P$ distinct from their intersection on one of their angle bisectors. The lines $f$ and $g$ passing through $P$ intersect the line $e_{i}$ at points $F_{i}$ and $G_{i}$ respectively. Determine the locus of the intersection of the lines $F_{1} G_{2}$ and $F_{2} G_{1}$.

Answer: -X


A plane cuts a sphere of radius  $ 1$  into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?

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


There are 100 lottery tickets. It is known that 5 tickets win 20 rubles, 10 tickets win 15 rubles, 15 tickets win 10 rubles, 25 tickets win 2 rubles, and the rest win nothing. Find the probability that a purchased ticket will win at least 10 rubles.

Answer: 0.30


Answer: \frac{3}{7}


Given \( x, y, z \geq 0 \) with at most one of them being zero, find the minimum value of
\[ f(x, y, z) = \sum \sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}} \]
where \(\sum\) represents a cyclic sum.

Answer: 12


In an isosceles trapezoid, the height is 10, and the diagonals are mutually perpendicular. Find the midsegment (the line connecting the midpoints of the non-parallel sides) of the trapezoid.

Answer: 10


Let \(a, b, c, d\) be real numbers such that \(c, d\) are not both 0. Define the function
\[
m(x)=\frac{ax+b}{cx+d}
\]
on all real numbers \(x\) except possibly \(-d/c\), in the event that \(c \neq 0\). Suppose that the equation \(x=m(m(x))\) has at least one solution that is not a solution of \(x=m(x)\). Find all possible values of \(a+d\). Prove that your answer is correct.

Answer: 0


The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD.

Answer: 25


The graph of the function \( f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) (where \( n > 1 \)), intersects the line \( y = b \) at the points \( B_1, B_2, \ldots, B_n \) (from left to right), and the line \( y = c \) (\( c \neq b \)) at the points \( C_1, C_2, \ldots, C_n \) (from left to right). Let \( P \) be a point on the line \( y = c \), to the right of the point \( C_n \). Find the sum \( \cot \angle B_1 C_1 P + \cdots + \cot \angle B_n C_n P \).

Answer: 0


We call a natural number interesting if all of its digits are different, and the sum of any two adjacent digits is a perfect square. Find the largest interesting number.

Answer: 6310972


An geometric sequence $\{a_n\}$ has 20 terms, where the product of the first four terms is $\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \_\_\_\_\_\_.

Answer: 32


The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$?

Answer: 16


Find all natural numbers \( x \) that satisfy the following conditions: the product of the digits of \( x \) is equal to \( 44x - 86868 \), and the sum of the digits is a cube of a natural number.

Answer: 1989


The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
 What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$?

Answer: 428


Under one of the cells on an $8 \times 8$ board, there is a hidden treasure. Under each of the other cells, there is a plaque that indicates the minimum number of steps needed to reach the treasure from that cell (one step allows moving to an adjacent cell by its side). What is the minimum number of cells that need to be dug up to certainly find the treasure?

Answer: 3


Let \( n \) be an integer of the form \( a^2 + b^2 \), where \( a \) and \( b \) are relatively prime integers and such that if \( p \) is a prime, \( p \leq \sqrt{n} \), then \( p \) divides \( a \) or \( b \). Determine all such \( n \).

Answer: 2,5,13


The rabbit and the turtle are traveling from point $A$ to the forest amusement park. The rabbit starts at 9:00 AM and jumps 40 meters in one minute, playing for 2 minutes after every 3 minutes of jumping. The turtle starts at 6:40 AM and crawls 10 meters per minute without any breaks. If the turtle arrives at the amusement park 15 seconds earlier than the rabbit, how many meters is it from point $A$ to the forest amusement park?

Answer: 2370


Given the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$.

Answer: -3


Prove that there exists an integer $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and such an $n$ is a multiple of 40.

Answer: 40


A right triangle has hypotenuse $c$, area $T$, and the radius of its inscribed circle is $\varrho$. Prove that the points of tangency of the inscribed circle form a triangle with area $\underline{\varrho}_{c} T$.

Answer: \frac{\varrho}{}T


For every integer  $n	\ge 2$  let  $B_n$  denote the set of all binary  $n$ -nuples of zeroes and ones, and split  $B_n$  into equivalence classes by letting two  $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers  $n 	\ge 2$  for which  $B_n$  splits into an odd number of equivalence classes.

Answer: 2


Using $\overline{a b c}$ and $\overline{d e f g}$ to represent a three-digit number and a four-digit number respectively, where $a, b, c, d, e, f, g$ each represent different digits from 1 to 9. Given that $\overline{a b c} + $\overline{d e f g} = 2020$, find the difference between the maximum and minimum values of $\overline{a b c} \times \overline{d e f g}$.

Answer: 552000


Given the function $f(x)=|2x+1|-|x|-2$.
(1) Solve the inequality $f(x)\geqslant 0$;
(2) If there exists a real number $x$ such that $f(x)-a\leqslant |x|$, find the minimum value of the real number $a$.

Answer: -3


Given  $a,x\in\mathbb{R}$  and  $x\geq 0$ , $a\geq 0$  . Also  $\sin(\sqrt{x+a})=\sin(\sqrt{x})$  . What can you say about  $a$ ??? Justify your answer.

Answer: 0


In a certain country, there are exactly 2019 cities and between any two of them, there is exactly one direct flight operated by an airline company, that is, given cities $A$ and $B$, there is either a flight from $A$ to $B$ or a flight from $B$ to $A$. Find the minimum number of airline companies operating in the country, knowing that direct flights between any three distinct cities are operated by different companies.

Answer: 2019


Let \( O \) be the center of the inscribed circle of triangle \( ABC \). Prove that \(\frac{OA^{2}}{bc}+\frac{OB^{2}}{ac}+\frac{OC^{2}}{ab}=1\).

Answer: 1


Given a tetrahedron \(ABCD\) with volume \(V\), the distances between three pairs of opposite edges \(AB\) and \(CD\), \(BC\) and \(AD\), \(AC\) and \(BD\) are \(d_1, d_2, d_3\) respectively, and the lengths of the lines connecting the midpoints of these pairs of opposite edges are \(m_1, m_2, m_3\) respectively, prove that \(d_1 d_2 d_3 \leq 3V \leq m_1 m_2 m_3\).

Answer: d_1d_2d_3\le3V\lem_1m_2m_3


A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".
    The probability that 10 minutes pass without either baby crying can be expressed as  $\frac{p}{q}$ . Compute  $p+q$ .

*2022 CCA Math Bonanza Team Round #7*

Answer: 337


The sequence of numbers $\left\{a_{n}\right\}$ is given by the conditions $a_{1}=1, a_{n+1}=\frac{3 a_{n}}{4}+\frac{1}{a_{n}}(n \geq 1)$.

Prove that:
a) The sequence $\left\{a_{n}\right\}$ is bounded.
b) $\left|a_{1000}-2\right|<2^{-1000}$.

Answer: |a_{1000}-2|<2^{-1000}


Given that \( I \) is the incenter of \( \triangle ABC \) and \( 5 \overrightarrow{IA} = 4(\overrightarrow{BI} + \overrightarrow{CI}) \). Let \( R \) and \( r \) be the radii of the circumcircle and the incircle of \( \triangle ABC \) respectively. If \( r = 15 \), then find \( R \).

Answer: 32


Given polynomials \( P(x) \) and \( Q(x) \), it is known that for some polynomial \( R(x, y) \), the equation
\[ P(x) - P(y) = R(x \cdot y)(Q(x) - Q(y)) \]
holds. Prove that there exists a polynomial \( S(x) \) such that \( P(x) = S(Q(x)) \).

Answer: P(x)=S(Q(x))


Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB.

Answer: 2\sqrt{3}


Given that 2019 satisfies the following properties:
(1) It is the sum of the fourth powers of five distinct positive integers:
$$
2019=1^{4}+2^{4}+3^{4}+5^{4}+6^{4};
$$
(2) It is the sum of six consecutive positive integers:
$$
2019=334+335+336+337+338+339.
$$

Prove that 2019 is the smallest positive integer that satisfies both properties simultaneously.

Answer: 2019


A sphere is centered at a point with integer coordinates and passes through the three points \((2,0,0)\), \((0,4,0)\), and \((0,0,6)\), but not the origin \((0,0,0)\). If \(r\) is the smallest possible radius of the sphere, compute \(r^{2}\).

Answer: 51


In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written?

Answer: 38


In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.

Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are in the line in total?

Answer: 12


Let \( p \) be a prime number and the sequence \(\{a_n\}_{n \geq 0}\) satisfy \( a_0 = 0 \), \( a_1 = 1 \), and for any non-negative integer \( n \), \( a_{n+2} = 2a_{n+1} - p a_n \). If \(-1\) is a term in the sequence \(\{a_n\}\), find all possible values of \( p \).

Answer: 5


Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of  $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

Answer: 33


From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that any two of the selected numbers do not include one number being seven times another. Find the maximum value of \(n\) and provide a justification.

Answer: 1763


Each of the two urns contains white and black balls, with the total number of balls in both urns equal to 25. One ball is drawn at random from each urn. Knowing that the probability of both drawn balls being white is 0.54, find the probability that both drawn balls will be black.

Answer: 0.04


At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$

Answer: \frac{1}{6}


Point \(A\) lies on the line \(y = \frac{15}{8}x - 8\), and point \(B\) lies on the parabola \(y = x^2\). What is the minimum length of segment \(AB\)?

Answer: 1823/544


Call a pair of integers  $(a,b)$  *primitive* if there exists a positive integer  $\ell$  such that  $(a+bi)^\ell$  is real. Find the smallest positive integer  $n$  such that less than  $1\%$  of the pairs  $(a, b)$  with  $0 \le a, b \le n$  are primitive.

*Proposed by Mehtaab Sawhney*

Answer: 299


What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time?

Answer: 2419


Answer: \frac{-21+\sqrt{33}}{2}


Fix positive integers  $k$  and  $n$ .Derive a simple expression involving Fibonacci numbers for the number of sequences  $(T_1,T_2,\ldots,T_k)$  of subsets  $T_i$  of  $[n]$  such that  $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$ .

Moderator says: and the original source for this one is Richard Stanley, *Enumerative Combinatorics* vol. 1 (1st edition), exercise 1.15.

Answer: F_{k+2}^n


The radius of the inscribed circle of a triangle is equal to 1. Prove that the smallest altitude of this triangle does not exceed 3.

Answer: 3


Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\overline{AG}$ and $\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?

Answer: 88/5


Diane has one 1-cent stamp, two identical 2-cent stamps, and so on, up to nine identical 9-cent stamps. In how many different arrangements can Diane paste exactly 10 cents worth of postage in a row across the top of an envelope? (Note, however, that simply rotating or inverting a stamp, or exchanging the positions of two stamps with the same denomination should be considered the same arrangement.)

Answer: 88


In which numeral system is the number \( 11111_{d} \) a perfect square?

Answer: 3


Answer: -7


Answer: 1


Teacher Xixi and teacher Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Teacher Xixi prepared a large bag of apples to distribute to her students, giving exactly 3 apples to each child; teacher Shanshan prepared a large bag of oranges to distribute to her students, giving exactly 5 oranges to each child. However, they mistakenly took each other's bags. In the end, teacher Xixi distributed 3 oranges to each child, but was short of 12 oranges; teacher Shanshan distributed 6 apples to each child, using up all the apples. How many apples did teacher Xixi prepare?

Answer: 72


A circle \(\omega\) is circumscribed around an acute-angled triangle \(ABC\). A point \(D\) is chosen on the side \(AB\), and a point \(E\) on the side \(BC\) such that \(AC \parallel DE\). Points \(P\) and \(Q\) on the smaller arc \(AC\) of the circle \(\omega\) are such that \(DP \parallel EQ\). The rays \(QA\) and \(PC\) intersect the line \(DE\) at points \(X\) and \(Y\) respectively. Prove that \(\angle XBY + \angle PBQ = 180^\circ\).

Answer: 180


Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is $10$. The area of $A 'B 'C'D'$ is

Answer: 50


In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter.

Answer: 15


$PS$  is a line segment of length  $4$  and  $O$  is the midpoint of  $PS$ . A semicircular arc is drawn with  $PS$  as diameter. Let  $X$  be the midpoint of this arc.  $Q$  and  $R$  are points on the arc  $PXS$  such that  $QR$  is parallel to  $PS$  and the semicircular arc drawn with  $QR$  as diameter is tangent to  $PS$ . What is the area of the region  $QXROQ$  bounded by the two semicircular arcs?

Answer: 2\pi-2


The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 points. In a game between player A and player B, the probability of the server scoring 1 point on each serve is 0.6, and the outcomes of each serve are independent of each other. Player A serves first in a game.
(1) Find the probability that the score is 1:2 in favor of player B at the start of the fourth serve;
(2) Find the probability that player A is leading in score at the start of the fifth serve.

Answer: 0.3072


There are 29 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 29 minutes?

Answer: 406


Three unit squares are given in the plane such that there is a point contained in all three. Furthermore, three points \( A, B, C \) are given, each contained in at least one of the squares. Find the maximum area of triangle \( ABC \).

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


In triangle \( \triangle ABC \), \(\angle ABC = 40^{\circ}\), \(\angle ACB = 20^{\circ}\), point \( N \) is inside \( \triangle ABC \), \(\angle NBC = 30^{\circ}\), and \(\angle NAB = 20^{\circ}\). Find the measure of \(\angle NCB\).

Answer: 10


There are 18 admission tickets to be distributed among four classes: Class A, Class B, Class C, and Class D. The requirements are that Class A must receive at least 1 ticket but no more than 5 tickets, Class B must receive at least 1 ticket but no more than 6 tickets, Class C must receive at least 2 tickets but no more than 7 tickets, and Class D must receive at least 4 tickets but no more than 10 tickets. How many different ways are there to distribute the tickets?

Answer: 140


Given two concentric circles and a point $A$ inside the smaller one. An angle of measure $\alpha$ with its vertex at $A$ cuts arcs on these circles. Prove that if the arc on the larger circle has an angular measure of $\alpha$, then the arc on the smaller circle also has an angular measure of $\alpha$.

Answer: \alpha


Given $\lg 2=0.3010$ and $\lg 3=0.4771$, at which decimal place does the first non-zero digit of $\left(\frac{6}{25}\right)^{100}$ occur?
(Shanghai Middle School Mathematics Competition, 1984)

Answer: 62


Let $(a,b,c,d)$ be a solution to the system\begin{align*}a+b&=15,\\ab+c+d&=78,\\ad+bc&=160,\\cd&=96.\end{align*}Find the greatest possible value of $a^2+b^2+c^2+d^2$.

Answer: 717


Given that \( a, b, c \) are the lengths of the sides of a right triangle, and for any natural number \( n > 2 \), the equation \(\left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2n} + b^{2n} + c^{2n}\right)\) holds, find \( n \).

Answer: 4


The line \(L\) contains the distinct points \(A\), \(B\), \(C\), \(D\) in that order. Construct a rectangle whose sides (or their extensions) intersect \(L\) at \(A\), \(B\), \(C\), and \(D\) such that the side which intersects \(L\) at \(C\) has length \(k\). How many such rectangles are there?

Answer: 2


Given that there are two types of golden triangles, one with a vertex angle of $36^{\circ}$ and the other with a vertex angle of $108^{\circ}$, and the ratio of the side opposite the $36^{\circ}$ angle to the side opposite the $72^{\circ}$ angle in the golden triangle with a vertex angle of $36^{\circ}$ is $\frac{\sqrt{5}-1}{2}$, express $\frac{1-2\sin^{2}27^{\circ}}{2t\sqrt{4-t^{2}}}$ in terms of $t$, where $t=\frac{\sqrt{5}-1}{2}$.

Answer: \frac{1}{4}


Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?

Answer: 10


We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace?

Answer: 18


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 a number by 3. How can one obtain the number 11 from the number 1 using this calculator?

Answer: 11


Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices.

Answer: 48


Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.

(I) Find the rectangular coordinate equation of curve $C\_2$ and the maximum value of the distance $|OP|$ between the moving point $P$ on curve $C\_1$ and the coordinate origin $O$;

(II) If curve $C\_2$ intersects with curve $C\_1$ at points $A$ and $B$, and intersects with the $x$-axis at point $E$, find the value of $|EA|+|EB|$.

Answer: \frac{6 \sqrt{3}}{5}


Let \( F(x) = |f(x) \cdot g(x)| \), where \( f(x) = a x^{2} + b x + c \) and \( g(x) = c x^{2} + b x + a \), with \( x \in [-1, 1] \). For any \( a, b, c \) such that \( |f(x)| \leq 1 \), find the maximum value of \( F(x) \) as \( a, b, \) and \( c \) vary.

Answer: 2


1. Player A writes down two rows of 10 positive integers, one under the other. The numbers must be chosen so that if \(a\) is under \(b\) and \(c\) is under \(d\), then \(a + d = b + c\). Player B is allowed to ask for the identity of the number in row \(i\), column \(j\). How many questions must he ask to be sure of determining all the numbers?

2. An \(m \times n\) array of positive integers is written on the blackboard. It has the property that for any four numbers \(a, b, c, d\) with \(a\) and \(b\) in the same row, \(c\) and \(d\) in the same row, \(a\) above \(c\) (in the same column) and \(b\) above \(d\) we have \(a + d = b + c\). If some numbers are wiped off, how many must be left for the table to be accurately restored?

Answer: +n-1


In every acyclic graph with 2022 vertices we can choose  $k$  of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of  $k$ .

Answer: 1517


Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.

Answer: 65


Answer: \log_{2008}(2009)


Let \(\mathbb{R}^{+}\) be the set of positive real numbers. Determine all functions \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) such that, for all positive real numbers \(x\) and \(y\),
$$
f(x+f(x y))+y=f(x) f(y)+1 .
$$

Answer: f(x)=x+1


Paul fills in a $7 \times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?

Answer: 147 / 2


From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is $\frac{2}{3}$ km from point $C$. Find the speed of the bus (in km/h), given that the distance between points $A$ and $B$ is 4 km and the speeds of the bus and the cyclist are constant.

Answer: 40


In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of the triangle is S, and it is given that 2$\sqrt {3}$S - $\overrightarrow {AB}$•$\overrightarrow {AC}$ = 0, and c = 2.
(I) Find the measure of angle A.
(II) If a² + b² - c² = $\frac {6}{5}$ab, find the value of b.

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


Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \).

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


Let set \( A = \{1, 2, \cdots, 366\} \). If a two-element subset \( B = \{a, b\} \) of \( A \) satisfies \( 17 \mid (a+b) \), then \( B \) is said to have property \( P \).

1. Find the number of two-element subsets of \( A \) that have property \( P \).
2. Find the number of two-element subsets of \( A \) that are pairwise disjoint and have property \( P \).

Answer: 179


Let \( 0 < x < \frac{\pi}{2} \). Determine the range of the function \( y = \sin x - 2 \cos x + \frac{32}{125 \sin x (1 - \cos x)} \).

Answer: [\frac{2}{5},+\infty)


Petya placed a ship of size $1 \times 4$ on a sea battle board (its size is $10 \times 10$). The ship can be placed both horizontally and vertically. Will Vasya be able to definitely hit it by making 24 shots?

Answer: Yes


Points $( \sqrt{\pi} , a)$ and $( \sqrt{\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)} \ \frac{\pi}{2} \qquad\textbf{(C)} \ 2 \qquad\textbf{(D)} \ \sqrt{1+\pi} \qquad\textbf{(E)} \ 1 + \sqrt{\pi}$

Answer: \textbf{(C)}\2


Five soccer teams held a tournament - each team played with each other once. For a win, 3 points were awarded, for a draw - 1 point, and no points were given for a loss. Four teams scored 1, 2, 5, and 7 points respectively. How many points did the fifth team score?

Answer: 12


Solve in  $\Bbb{N}^*$  the equation   $$  4^a \cdot 5^b - 3^c \cdot 11^d = 1. $$

Answer: (,b,,)=(1,2,2,1)


Triangle $A B C$ has $A B=4, B C=5$, and $C A=6$. Points $A^{\prime}, B^{\prime}, C^{\prime}$ are such that $B^{\prime} C^{\prime}$ is tangent to the circumcircle of $\triangle A B C$ at $A, C^{\prime} A^{\prime}$ is tangent to the circumcircle at $B$, and $A^{\prime} B^{\prime}$ is tangent to the circumcircle at $C$. Find the length $B^{\prime} C^{\prime}$.

Answer: \frac{80}{3}


For how many real numbers $x$ is $\sqrt{-(x+1)^2}$ a real number?
$\textbf{(A) }\text{none}\qquad \textbf{(B) }\text{one}\qquad \textbf{(C) }\text{two}\qquad\\ \textbf{(D) }\text{a finite number greater than two}\qquad \textbf{(E) }\infty$

Answer: \textbf{(B)}


Determine the largest integer \( n \) such that \( 7^{2048} - 1 \) is divisible by \( 2^{n} \).

Answer: 14


The domain of the function \( y = f(x) \) is \( D \). If, for any \( x_1 \in D \), there exists a unique \( x_2 \in D \) such that \( f(x_1) + f(x_2) = 2008 \) holds, then \( x_2 \) is called the "Olympic number" of \( x_1 \). Given the following five functions:
1. \( y = 2008 x^2 \)
2. \( y = 2008 x^3 \)
3. \( y = 2008 \cos x \)
4. \( y = \lg(2008x) \)
5. \( y = 2008^x \)

Identify which of these functions have the "Olympic number" property. Fill in the numbers of the functions you think meet the requirement.

Answer: (2),(4)


Points \( A, B, C, D \) lie on the circumference of a circle, and \( BC = DC = 4 \), \( AE = 6 \). The lengths of segments \( BE \) and \( DE \) are both integers. Find the length of \( BD \).

Answer: 7


a and b are real numbers for which the equation \(x^4 + ax^3 + bx^2 + ax + 1 = 0\) has at least one real solution. Find the least possible value of \(a^2 + b^2\).

Answer: 4/5


Find the number of three-digit numbers which are multiples of 3 and are formed by the digits 0, 1, 2, 3, 4, 5, 6, and 7 without repetition.

Answer: 106


Let  $p$  be a prime number. All natural numbers from  $1$  to  $p$  are written in a row in ascending order. Find all  $p$  such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.

*A. Khrabov*

Answer: 3


Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
$\textbf{(A) }1\dfrac{1}{4}\qquad\textbf{(B) }3\dfrac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25$

Answer: \textbf{(D)}\5


Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $\overrightarrow{m} = \overrightarrow{a} + t \overrightarrow{b} (t \in \mathbb{R})$, $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, the minimum value of $|\overrightarrow{m}|$ is obtained only when $t = \frac{1}{4}$. Determine the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


An express train from Basltown to Ironchester travels at a speed of 60 km/h, while an express train from Ironchester to Basltown, which departs at the same time, travels at a speed of 40 km/h.

What distance will they be apart one hour before their meeting point?

Let's assume the total distance between the two cities does not exceed 250 km.

Answer: 100\,


A triangle with side lengths $5,7,8$ is inscribed in a circle $C$. The diameters of $C$ parallel to the sides of lengths 5 and 8 divide $C$ into four sectors. What is the area of either of the two smaller ones?

Answer: \frac{49}{18} \pi


There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off?

Answer: 20


Let \( S = \{1, 2, 3, 4\} \). A sequence of \( n \) terms \( a_1, a_2, \cdots, a_n \) has the following property: for any non-empty subset \( B \) of \( S \) (denote the number of elements in \( B \) as \( |B| \)), there exist \( |B| \) adjacent terms in the sequence that exactly form the subset \( B \). Find the minimum value of \( n \).

Answer: 8


Three fair six-sided dice, each numbered 1 through 6 , are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?

Answer: 37/72


Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - 4f(x) = 0 \) has exactly three solutions. How many solutions does the equation \( (f(x))^2 = 1 \) have?

Answer: 2


Answer: 300,500,000


Let  $ABC$  be an acute, non-isosceles triangle which is inscribed in a circle  $(O)$ . A point  $I$  belongs to the segment  $BC$ . Denote by  $H$  and  $K$  the projections of  $I$  on  $AB$  and  $AC$ , respectively. Suppose that the line  $HK $  intersects  $(O)$  at  $M, N$  ( $H$  is between  $M, K$  and  $K$  is between  $H, N$ ). Let  $X, Y$  be the centers of the circles  $(ABK),(ACH)$  respectively. Prove the
following assertions:
a) If  $I$  is the projection of  $A$  on  $BC$ , then  $A$  is the center of circle  $(IMN)$ .
b) If  $XY\parallel BC$ , then the orthocenter of  $XOY$  is the midpoint of  $IO$ .

Answer: Theorthocenterof\triangleXOYisthemidpointofIO


Answer: 0


In triangle \(ABC\), \( M \) is the midpoint of the arc \(ABC\) of the circumcircle of the triangle, \(NP\) and \(NT\) are tangents to the incircle. The lines \(BP\) and \(BT\) intersect the circumcircle of the triangle again at points \(P_1\) and \(T_1\) respectively. Prove that \(PP_1 = TT_1\).

Answer: PP_1=TT_1


Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______.

Answer: \frac{\pi}{6}


There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
 Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
[asy] pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/asy]

Answer: 336


The differentiable function \( f:\left(0, \frac{\pi}{2}\right) \rightarrow \mathbf{R} \) satisfies \( f'(x) \tan x \geqslant (2 \sin x - 1) f(x) \). Given that \( f\left(\frac{\pi}{6}\right) = 1 \), find the minimum value of \( f\left(\frac{\pi}{3}\right) \).

Answer: \frac{e^{\sqrt{3}-1}}{\sqrt{3}}


A four-digit integer has its digits increasing from left to right. When we reverse the order of the digits, we obtain a four-digit integer whose digits decrease from left to right. A third four-digit integer uses exactly the same digits, but in a different order. The sum of the three integers is 26352. What is the value of the smallest of the three integers?

Answer: 6789


In the cube \( ABCD-A_1B_1C_1D_1 \), \( E \) is the midpoint of \( BC \), and \( F \) lies on \( AA_1 \) such that \( A_1F : FA = 1 : 2 \). Find the dihedral angle between plane \( B_1EF \) and the base \( A_1B_1C_1D_1 \).

Answer: \arctan\frac{\sqrt{37}}{3}