Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$  Find the smallest possible value of $a.$

Answer: 10 \pi - 29


Two circles touch each other internally. It is known that two radii of the larger circle, which form an angle of $60^\circ$ between them, are tangent to the smaller circle. Find the ratio of the radii of the circles.

Answer: 3


Bob writes a random string of 5 letters, where each letter is either $A, B, C$, or $D$. The letter in each position is independently chosen, and each of the letters $A, B, C, D$ is chosen with equal probability. Given that there are at least two $A$ 's in the string, find the probability that there are at least three $A$ 's in the string.

Answer: 53/188


Find all values of \( a \) for which the system

\[
\left\{
\begin{array}{l}
x^{2} + 4y^{2} = 1 \\
x + 2y = a
\end{array}
\right.
\]

has a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0.

Answer: -1.41


Let $\bigtriangleup PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a, b, c,$ and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a$ and $d$ are relatively prime, and c is not divisible by the square of any prime. Find $a+b+c+d$.

Answer: 21


Let \( f(x) \) be a function defined for all real numbers \( x \neq 0 \) such that for any non-zero real numbers \( x_1 \) and \( x_2 \), the equation \( f(x_1 x_2) = f(x_1) + f(x_2) \) holds. Additionally, \( f(x) \) is an increasing function on the interval \( (0, +\infty) \). Determine the solution set for the inequality \( f(x) + f\left(x - \frac{1}{2}\right) \leqslant 0 \).

Answer: [\frac{1-\sqrt{17}}{4},0)\cup(0,\frac{1}{2})\cup(\frac{1}{2},\frac{1+\sqrt{17}}{4}]


Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens' digits are different from each others' tens' digits and all of their numbers' ones digits are different from each others' ones' digits?

Answer: \frac{112}{225}


$39$  students participated in a math competition. The exam consisted of  $6$  problems and each problem was worth  $1$  point for a correct solution and  $0$  points for an incorrect solution. For any  $3$  students, there is at most  $1$  problem that was not solved by any of the three. Let  $B$  be the sum of all of the scores of the  $39$  students. Find the smallest possible value of  $B$ .

Answer: 165


In triangle $XYZ$, $XY = 540$ and $YZ = 360$. Points $N$ and $O$ are located on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XN = NY$, and $\overline{ZO}$ is the angle bisector of angle $Z$. Let $Q$ be the point of intersection of $\overline{YN}$ and $\overline{ZO}$, and let $R$ be the point on line $YN$ for which $N$ is the midpoint of $\overline{RQ}$. If $XR = 216$, find $OQ$.

Answer: 216


Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.

Answer: 120


Write 22 as the sum of several distinct natural numbers so that the product of these numbers is maximized. The maximum product is \_\_\_\_\_\_.

Answer: 1008


Let  $m$  be an integer where  $|m|\ge 2$ . Let  $a_1,a_2,\cdots$  be a sequence of integers such that  $a_1,a_2$  are not both zero, and for any positive integer  $n$ ,  $a_{n+2}=a_{n+1}-ma_n$ .

Prove that if positive integers  $r>s\ge 2$  satisfy  $a_r=a_s=a_1$ , then  $r-s\ge |m|$ .

Answer: r-\ge||


The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one?

Answer: 124


Given a right isosceles triangle \(ABC\) with the right angle at \(C\) and legs of length 2. An arc of a circle \(l\) centered at \(A\) divides the triangle into two equal areas, while an arc of a circle \(m\) centered at \(B\) is tangent to arc \(l\) at a point on the hypotenuse \(AB\).

Find the area of the portion of the triangle not covered by the circular sectors corresponding to the two arcs.

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


Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday?
$\mathrm{(A)}\ 2011 \qquad \mathrm{(B)}\ 2012 \qquad \mathrm{(C)}\ 2013 \qquad \mathrm{(D)}\ 2015 \qquad \mathrm{(E)}\ 2017$

Answer: (E)\2017


The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.

How many fruits in total grew on the miracle tree?

Answer: 30


Answer: 924


Within the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, there is a point $M(P, 0)$ where $0 < P < a$. A moving point $Q$ on the ellipse is such that the vector $\overrightarrow{MQ}$ is rotated $90^\circ$ clockwise around point $M$ to become $\overrightarrow{MP}$. Find the equation of the locus of point $P$.

Answer: \frac{(x-P)^2}{b^2}+\frac{(y-P)^2}{^2}=1


In a right triangular prism $\mathrm{ABC}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$, the lengths of the base edges and the lateral edges are all 2. If $\mathrm{E}$ is the midpoint of $\mathrm{CC}_{1}$, what is the distance from $\mathrm{C}_{1}$ to the plane $\mathrm{AB} \mathrm{B}_{1} \mathrm{E}$?

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


Let \( S = \{1, 2, \cdots, 2016\} \). For any non-empty finite sets of real numbers \( A \) and \( B \), find the minimum value of
\[ f = |A \Delta S| + |B \Delta S| + |C \Delta S| \]
where
\[ X \Delta Y = \{a \in X \mid a \notin Y\} \cup \{a \in Y \mid a \notin X\} \]
is the symmetric difference between sets \( X \) and \( Y \), and
\[ C = \{a + b \mid a \in A, b \in B\} .\]

Answer: 2017


What is the sum of the positive solutions to \( 2x^2 - x \lfloor x \rfloor = 5 \), where \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \)?

Answer: \frac{3 + \sqrt{41} + 2\sqrt{11}}{4}


2016 students are lined up in a row and count off from left to right according to $1,2 \cdots, n$ $(n \geqslant 2)$. If the 2016th student counts to $n$, all students who count to $n$ in this round will receive a New Year's gift. How many students will definitely not receive a New Year's gift, regardless of what $n$ is?

Answer: 576


Answer: 2000


Bob, a bomb-defuser, encounters a bomb with two wires: red and green. Initially, he decides to cut the green wire. However, every second from time \( t = 15 \) seconds down to \( t = 1 \) second, there is a \(\frac{1}{2t^2}\) chance that Bob, due to panic, will switch his wirecutters to the other wire. At \( t = 1 \) second, Bob will cut the wire his wirecutters are currently on without switching. What is the probability that Bob cuts the green wire?

Answer: \frac{23}{30}


Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the 2017 integers that began in the list, how many are now missing?

Answer: 373


Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \]

Answer: 2043231


Given that \(\alpha\) and \(\beta\) are acute angles, and the following equations hold:
$$
\left\{\begin{array}{l}
3 \sin ^{2} \alpha + 2 \sin ^{2} \beta = 1, \\
3 \sin 2 \alpha - 2 \sin 2 \beta = 0.
\end{array}\right.
$$

Determine \(\alpha + 2\beta\).

Answer: \frac{\pi}{2}


A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?

Answer: 4


There was a solution in a glass, consisting of 99% water. The glass with the solution was weighed and found to weigh 500 grams. After some water evaporated, the water content became 98%. How much will the glass with the resulting solution weigh if the empty glass weighs 300 grams?

Answer: 400\,


Set \( A = \{1, 2, \cdots, n\} \). If there exist nonempty sets \( B \) and \( C \) such that \( B \cap C = \emptyset \), \( B \cup C = A \), and the sum of the squares of the elements in \( B \) is \( M \), and the sum of the squares of the elements in \( C \) is \( N \), and \( M - N = 2016 \), find the smallest value of \( n \).

Answer: 19


Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number.

Answer: 30


The kindergarten teacher evenly distributed 270 apples, 180 pears, and 235 oranges to the larger group of children. The remaining quantities of apples, pears, and oranges are in the ratio $3:2:1$. How many children are there in the larger group?

Answer: 29


Given vectors $\overrightarrow{a}=(1, -2)$ and $\overrightarrow{b}=(3, 4)$, the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$ is ______.

Answer: -1


Given the set of natural numbers \(\mathbf{N}\), the set of real numbers \(\mathbf{R}\), and the set \(S\) of functions \(f: \mathbf{N} \rightarrow \mathbf{R}\) that satisfy the following two conditions:
1. \(f(1) = 2\)
2. \(f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n)\) for \(n = 1, 2, \cdots\)

Find the smallest natural number \(M\) such that for any \(f \in S\) and any \(n \in \mathbf{N}\), \(f(n) < M\).

Answer: 10


On the circumference of the base of a right circular cylinder, select diametrically opposite points $A$ and $B$. On the circumference of the other base, select a point $C$ such that $C$ is not in the plane $ABO$, where $O$ is the midpoint of the cylinder's axis. Prove that for the trihedral angle with vertex $O$ and edges $OA$, $OB$, and $OC$, the sum of the pairwise dihedral angles amounts to $360^\circ$.

Answer: 360


The incenter of the triangle  $ ABC$  is  $ K.$  The midpoint of  $ AB$  is  $ C_1$  and that of  $ AC$  is  $ B_1.$  The lines  $ C_1K$  and  $ AC$  meet at  $ B_2,$  the lines  $ B_1K$  and  $ AB$  at  $ C_2.$  If the areas of the triangles  $ AB_2C_2$  and  $ ABC$  are equal, what is the measure of angle  $ \angle CAB?$

Answer: 60


At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make?

Answer: 126


Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves.

Answer: \dfrac{\sqrt{3}}{3}


Given the sequence \(\{a_n\}\) satisfies

\[ a_n + a_{n+1} = n(-1)^{\frac{n(n+1)}{2}}, \]

and the sum of the first \( n \) terms is \( S_n \). Also, \( m + S_{2015} = -1007 \) and \( a_1 m > 0 \).

Find the minimum value of \( \frac{1}{a_1} + \frac{4}{m} \).

Answer: 9


A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original list price should the marked price be?

Answer: 107\%


What is $2\frac{1}{4}$ divided by $\frac{2}{3}$?

Answer: \frac{27}{8}


Each face of a hexahedron and each face of a regular octahedron are squares with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is an irreducible fraction $\frac{m}{n}$. What is the product $m \cdot n$?

Answer: 6


Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\] can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Answer: 289


Given the family of curves \(2(2 \sin \theta - \cos \theta + 3)x^2 - (8 \sin \theta + \cos \theta + 1)y = 0\), where \(\theta\) is a parameter, find the maximum length of the chord intercepted by these curves on the line \(y = 2x\).

Answer: 8\sqrt{5}


Find the flux of the vector field **a** = (x - 2z)i + (x + 3y + z)j + (5x + y)k through the upper side of the triangle ABC with vertices at points A(1,0,0), B(0,1,0), C(0,0,1).

Answer: \frac{5}{3}


Let  $ABC$  be a fixed triangle in the plane. Let  $D$  be an arbitrary point in the plane. The circle with center  $D$ , passing through  $A$ , meets  $AB$  and  $AC$  again at points  $A_b$  and  $A_c$  respectively. Points  $B_a, B_c, C_a$  and  $C_b$  are defined similarly. A point  $D$  is called good if the points  $A_b, A_c,B_a, B_c, C_a$ , and  $C_b$  are concyclic. For a given triangle  $ABC$ , how many good points can there be?

(A. Garkavyj, A. Sokolov )

Answer: 4


Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$.

Answer: -1


Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a door and tries to put them on a random one of his cat's legs; however, Tac never tries to put more than one of each type of clothing on each leg of his cat. What is the probability that, after Tac is done, the snow-shoe on each of his cat's legs is on top of the shoe, which is on top of the sock?

Answer: $\frac{1}{1296}$


Prove that \( 20Rr - 4r^2 \leq ab + bc + ca \leq 4(R + r)^2 \).

Answer: 20Rr-4r^{2}\le++ca\le4(R+r)^2


In triangle \( PQR \), the angle \( \angle QPR \) is \( 60^{\circ} \). Perpendiculars are drawn from vertices \( P \) and \( R \) to the sides \( QR \) and \( PQ \) respectively. The point of intersection of these perpendiculars is 1 unit away from vertices \( P \) and \( Q \). Find the sides of triangle \( PQR \).

Answer: \sqrt{3}


Xiao Qian, Xiao Lu, and Xiao Dai are guessing a natural number between $1$ and $99$. Here are their statements:

Xiao Qian says: "It is a perfect square number, and it is less than 5."
Xiao Lu says: "It is less than 7, and it is a two-digit number."
Xiao Dai says: "The first half of Xiao Qian’s statement is true, but the second half is false."

If among these 3 people, one person’s two statements are both true, one person’s two statements are both false, and one person’s one statement is true and the other is false, then what is the number? (Note: A perfect square number is a number that can be expressed as the square of an integer, for example, $4=2^2$, $81=9^2$, so we call 4, 9 perfect square numbers).

Answer: 9


Find the minimum value of the expression:

\[
\left(\sqrt{2(1+\cos 2x)} - \sqrt{9-\sqrt{7}} \sin x + 1\right) \cdot \left(3 + 2 \sqrt{13 - \sqrt{7}} \cos y - \cos 2y\right)
\]

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

Answer: -19


Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist?

Answer: 33


Let \( C_1 \) and \( C_2 \) be two concentric circles, with \( C_2 \) inside \( C_1 \). Let \( A \) be a point on \( C_1 \) and \( B \) a point on \( C_2 \) such that the line segment \( AB \) is tangent to \( C_2 \). Let \( C \) be the second point of intersection of the line segment \( AB \) with \( C_1 \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( C_2 \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) and \( CF \) intersect at a point \( M \) on the line segment \( AB \). Determine the ratio \( AM / MC \).

Answer: 1


Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime.

Answer: 41


Find the number of matrices satisfying two conditions:
1) The matrix has the form \(\left(\begin{array}{lll}1 & * & * \\ * & 1 & * \\ * & * & 1\end{array}\right)\), where each * can take the value 0 or 1.
2) The rows of the matrix do not repeat.

Answer: 42


If a collection of lines in a plane has no two lines parallel and no three lines concurrent, the collection is said to be in "general position." A collection of lines in general position divides the plane into various regions, with the bounded regions referred to as the "finite regions" of the collection. Prove that for sufficiently large \( n \) and any collection of \( n \) lines in general position, at least \( \frac{2 \sqrt{n}}{3} \) of these lines can be colored blue so that the boundary of every finite region is not entirely blue.

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


Consider the set \( M \) consisting of 100 points in the Cartesian plane. Prove that there are at most 2025 rectangles whose vertices are all in \( M \) and whose sides are parallel to the coordinate axes.

Answer: 2025


Given the function $f(x) = |x^2 + bx|$ ($b \in \mathbb{R}$), when $x \in [0, 1]$, find the minimum value of the maximum value of $f(x)$.

Answer: 3-2\sqrt{2}


On a certain unidirectional highway, trucks move steadily at 60 miles per hour spaced \( \frac{1}{4} \) of a mile apart. Cars move steadily at 75 miles per hour spaced 3 seconds apart. A lone sports car weaving through traffic at a steady forward speed passes two cars between each truck it passes. How quickly is it moving in miles per hour?

Answer: 90\,


Determine the number of distinct values of the expression

$$
\frac{n^{2}-2}{n^{2}-n+2}
$$

where \( n \in \{1, 2, \ldots, 100\} \).

Answer: 98


Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability  $\frac{1}{2}$ . What is the expected value of the number of games they will play?

Answer: 14


Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), and \( M \) and \( I \) are the centroid and incenter of \(\triangle P F_{1} F_{2}\) respectively, if \( M I \) is perpendicular to the \( x \)-axis, then the radius of the incircle of \(\triangle P F_{1} F_{2}\) is _____.

Answer: \sqrt{6}


The rhombus  $ABCD$  is given. Let  $E$  be one of the points of intersection of the circles  $\Gamma_B$  and  $\Gamma_C$ , where  $\Gamma_B$  is the circle centered at  $B$  and passing through  $C$ , and  $\Gamma_C$  is the circle centered at  $C$  and passing through  $B$ . The line  $ED$  intersects  $\Gamma_B$  at point  $F$ .
Find the value of angle  $\angle AFB$ .

*(S. Mazanik)*

Answer: 60


Does there exist a number with 2020 digits (not containing the digits 0 and 1) that is divisible by the sum of its digits?

Answer: Yes


The average years of experience of three employees, David, Emma, and Fiona, at a company is 12 years. Five years ago, Fiona had the same years of experience as David has now. In 4 years, Emma's experience will be $\frac{3}{4}$ of David's experience at that time. How many years of experience does Fiona have now?

Answer: \frac{183}{11}


Along the side of an equilateral triangle, a circle with a radius equal to the height of the triangle rolls. Prove that the angular measure of the arc intercepted on the circle by the sides of the triangle is always $60^{0}$.

Answer: 60


How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges.

Answer: 11


Find the number of pairs of integers \( (x, y) \) that satisfy the condition \( x^{2} + 6xy + 5y^{2} = 10^{100} \).

Answer: 19594


An infinite sheet of paper is divided into equal squares, some of which are colored red. In each  $2\times3$  rectangle, there are exactly two red squares. Now consider an arbitrary  $9\times11$  rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

Answer: 33


Find all triples of positive integers $(x,y,z)$ that satisfy the equation
$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$

Answer: (2, 3, 3)


Leticia has a  $9\times 9$  board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has  $4$  friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:

 - If the square is green, write the number of red friends plus twice the number of blue friends.

 - If the square is red, write the number of blue friends plus twice the number of green friends.

 - If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Answer: 486


Find the values of  $a\in [0,\infty)$  for which there exist continuous functions  $f:\mathbb{R}\rightarrow \mathbb{R}$ , such that  $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$ .

Answer: 0


In  $10\times 10$  square we choose  $n$  cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of  $n$ ?

Answer: 50


In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?

Answer: 117


The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\le i \le 7$ , so that the sum of the $A_i$ 's is also $19$ and so that $M$ , the maximum of the "errors" $\lvert A_i-a_i \rvert$ , is as small as possible. For this minimum $M$ , what is $100M$ ?

Answer: 61


The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?

Answer: 75


Given that \( s \) and \( t \) are positive integers, and \( T \) is a tree of order \( t \), then \( R(T, K^s) = (s-1)(t-1) + 1 \).

Answer: (-1)(-1)+1


In a wooden square with a grid pattern, 102 cells are painted black. Petya used this square as a stamp and pressed it onto a white sheet of paper 100 times. Each time, these 102 cells (and only these) left a black mark on the paper. Could it be possible that, in the end, on the sheet, there results a $101 \times 101$ square where every cell, except for one corner cell, is black?

Answer: Yes


Given a real number \(C\) and a sequence of real numbers \(\{a_n\}\). For each positive integer \(n\), let \(M_n = \frac{a_1 + a_2 + \cdots + a_n}{n}\). It is given that for all distinct positive integers \(i, j, k\), the following holds:
$$
(i-j) M_k + (j-k) M_i + (k-i) M_j = C.
$$

Prove that \(\{a_n\}\) is an arithmetic sequence.

Answer: Thesequence{a_n}isanarithmeticsequence.


Given the function $f(x)=ax + a^{-x}$ ($a>0$ and $a\neq1$), and $f(1)=3$, find the value of $f(0)+f(1)+f(2)$.

Answer: 12


Do there exist real numbers \( a, b, \) and \( c \) such that for all real numbers \( x \) and \( y \), the inequality

\[
|x+a| + |x+y+b| + |y+c| > |x| + |x+y| + |y|
\]

holds?

Answer: No


In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the internal angles $A$, $B$, and $C$, respectively, and $\sin \left(A-B\right)\cos C=\cos B\sin \left(A-C\right)$.
$(1)$ Determine the shape of triangle $\triangle ABC$;
$(2)$ If triangle $\triangle ABC$ is an acute triangle and $a=\frac{1}{\sin B}$, find the maximum value of $\frac{{b}^{2}+{a}^{2}}{(ab)^{2}}$.

Answer: \frac{25}{16}


Given that the vertex of angle $\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\frac{3}{5}, -\frac{4}{5})$.

(1) Find the value of $\sin(\alpha + \pi)$;

(2) If angle $\beta$ satisfies $\sin(\alpha + \beta) = \frac{5}{13}$, find the value of $\cos(\beta)$.

Answer: \frac{16}{65}


The coefficients of the polynomial \(P(x)\) are nonnegative integers, each less than 100. Given that \(P(10)=331633\) and \(P(-10)=273373\), compute \(P(1)\).

Answer: 100


Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is \(\frac{1}{2}\). After she wins a game, the probability of Carina winning the next game is \(\frac{3}{4}\). After she loses a game, the probability of Carina winning the next game is \(\frac{1}{3}\). The probability that Carina wins 3 games before being eliminated from the tournament equals \(\frac{a}{b}\), where the fraction \(\frac{a}{b}\) is in lowest terms. What is the value of \(a+b\)?

Answer: 23


In triangle \( ABC \), the midpoints of sides \( AC \) and \( BC \) are marked as points \( M \) and \( N \) respectively. The angle \( MAN \) is \( 15^{\circ} \), and the angle \( BAN \) is \( 45^{\circ} \).

Find the angle \( ABM \).

Answer: 75


Given the hyperbola with the equation $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$, where $F\_1$ and $F\_2$ are its foci, and point $M$ lies on the hyperbola.
(1) If $\angle F\_1 M F\_2 = 90^{\circ}$, find the area of $\triangle F\_1 M F\_2$.
(2) If $\angle F\_1 M F\_2 = 60^{\circ}$, what is the area of $\triangle F\_1 M F\_2$? If $\angle F\_1 M F\_2 = 120^{\circ}$, what is the area of $\triangle F\_1 M F\_2$?

Answer: 3 \sqrt{3}


Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \).

Answer: 2^{501}


Let \( O \) be the vertex of the parabola and \( F \) be the focus, and let \( PQ \) be a chord passing through \( F \). Given \( |OF| = a \) and \( |PQ| = b \), find the area of triangle \( OPQ \).

Answer: \sqrt{}


Find the number of permutations  $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$  of  $1, 2 ,3,4,5,6$  such that for any  $k, 1 \leq k \leq 5$ ,  $(p_1, \ldots, p_k)$  does not form a permutation of  $1 , 2, \ldots, k$ .

Answer: 461


On a planet, there are $2^{N}$ countries $(N \geqslant 5)$. Each country has a flag consisting of a row of $N$ squares of side 1, each being either yellow or blue. The flags are pairwise distinct. A set of $N$ flags is diverse if they can be arranged in a certain order to form an $N \times N$ square such that the $N$ squares on the main diagonal are of the same color. Determine the smallest integer $M$ such that every set of $M$ flags contains a diverse set of $N$ flags.

Answer: 2^{N-2}+1


A teacher wanted to write an arithmetic problem on the board:

\[ 1.05 + 1.15 + 1.25 + 1.4 + 1.5 + 1.6 + 1.75 + 1.85 + 1.95 = ? \]

but accidentally forgot to write one decimal point. After that, Kolya went to the board, correctly performed all the calculations, and obtained an integer as the result. What is that integer?

Answer: 27


\(ABCD\) is a square with side length 9. Let \(P\) be a point on \(AB\) such that \(AP: PB = 7:2\). Using \(C\) as the center and \(CB\) as the radius, a quarter circle is drawn inside the square. The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). The segments \(CE\) and \(DB\) meet at \(K\), while \(AK\) and \(PQ\) meet at \(M\). Find the length of \(AM\).

Answer: \frac{85}{22}


Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$?

Answer: \frac{1}{32}


Calculate

$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2019 \pi}{43} \cdot \operatorname{tg} \frac{2020 \pi}{43}
$$

Answer: -2021


An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment?

Answer: 13


Given a positive integer  $m$ , let  $d(m)$  be the number of positive divisors of  $m$ . Determine all positive integers  $n$  such that  $d(n) +d(n+ 1) = 5$ .

Answer: 3,4


What is the largest possible area of the orthogonal projection of a $3 \times 4 \times 12$ rectangular prism?

Answer: 12\sqrt{26}


A and B are running back and forth on a 120-meter straight road. A's speed is 5 meters per second, and B's speed is 3 meters per second. If they start running from the same end at the same time and run for 15 minutes, how many times do they meet head-on during this period (excluding the endpoints)?

Answer: 23


Given the real number \( x \) satisfies \( 20 \sin x = 22 \cos x \), find the largest integer not greater than the real number \( \left( \frac{1}{\sin x \cos x} - 1 \right)^7 \).

Answer: 1


Gavrila got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was fully discharged. He spent half of the total time playing Tetris and the other half watching cartoons. It is known that the smartphone completely discharges in 3 hours of video playback or 5 hours of playing Tetris. What distance did Gavrila travel if the train moved at an average speed of 80 km/h for half the way and at an average speed of 60 km/h for the other half? Give the answer in kilometers, rounded to the nearest whole number if necessary.

Answer: 257


A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is at the back, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the truck; 5 minutes later, the car catches up with the bus; another $t$ minutes later, the truck catches up with the bus. Find the value of $t$.

Answer: 15


Let  $a$ ,  $b$ ,  $c$  be the sides of an acute triangle  $\triangle ABC$  , then for any  $x, y, z \geq 0$ , such that  $xy+yz+zx=1$  holds inequality:  $$ a^2x + b^2y + c^2z \geq 4F $$   where  $F$  is the area of the triangle  $\triangle ABC$

Answer: ^2x+b^2y+^2z\ge4F


Find the volume of a regular hexagonal pyramid with a base side length \( a \) and the radius \( R \) of the circumscribed sphere.

Answer: \frac{\sqrt{3}}{2}^2(R+\sqrt{R^2-^2})


A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$.

Answer: 3\sqrt{2}


On the edge $BB_{1}$ of a regular triangular prism $ABC A_{1} B_{1} C_{1}$, a point $T$ is taken such that $BT: B_{1} T=2:3$. Point $T$ is the vertex of a right circular cone, such that three vertices of the prism belong to the circle of its base.

a) Find the ratio of the height of the prism to the edge of its base.

b) Let it be additionally known that $C C_{1}=5$. Find the volume of the cone.

Answer: \frac{180\pi\sqrt{3}}{23\sqrt{23}}


If the number of sides of a polygon is doubled, the number of diagonals increases by 45. What is this polygon?

Answer: 6


Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).
$A$: $x_1^2x_2^2-x_1-x_2=115$;
$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.
$(1)$ Find the value of $k$;
$(2)$ Solve this equation.

Answer: -11


Given the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1\) and a point \((\sqrt{5}, \sqrt{2})\) inside the ellipse, draw two chords \(AB\) and \(CD\). Tangents to the ellipse at points \(A\) and \(B\) intersect at point \(E\), and tangents to the ellipse at points \(C\) and \(D\) intersect at point \(F\). Find the equation of the line \(EF\).

Answer: \frac{\sqrt{5}}{9}x+\frac{\sqrt{2}}{5}y=1


From the sixteen natural numbers $1, 2, 3, 4, \cdots, 15, 16$, if $n$ numbers are randomly selected, there will always be such two numbers: one is three times the other. What is the smallest $n$?

Answer: 13


Given that $\alpha, \beta, \gamma$ are acute angles, $\operatorname{tg} \frac{\alpha}{2} = \operatorname{tg}^{3} \frac{\gamma}{2}$, and $2 \operatorname{tg} \beta = \operatorname{tg} \gamma$. Prove that $\alpha, \beta, \gamma$ form an arithmetic sequence.

Answer: \alpha,\beta,\gamma


Several resistances are connected between terminals $A$ and $B$. Each resistance has an input and an output terminal. What is the minimum number of resistances needed and what could be their connection scheme, so that if any nine resistances fail (either a short circuit or an open circuit), the circuit still connects terminals $A$ and $B$ without causing a short circuit?

Answer: 100


Cyclists Alexei, Vitaly, and Sergei participated in a 10 km race. At the moment Alexei finished, Vitaly had 1 km left to ride. When Vitaly finished, Sergei had 1 km left to ride. By how many kilometers did Alexei finish ahead of Sergei?

Answer: 1.9


Define an ordered quadruple of integers  $(a, b, c, d)$  as interesting if  $1 \le a<b<c<d \le 10$ , and  $a+d>b+c$ . How many ordered quadruples are there?

Answer: 80


Carl and Dina each arrive at a café at a random time between 1:00 PM and 1:45 PM. Each stays for 30 minutes before leaving. What is the probability that Carl and Dina meet each other at the café?

Answer: \frac{8}{9}


Let \( A_{1}, A_{2}, \cdots, A_{n} \) be \( r \)-element sets such that \( \bigcup_{i=1}^{n} A_{i} = X \). If for any natural number \( k \), the union of any \( k \) of these sets is \( X \), and the union of any \( k-1 \) of these sets is a proper subset of \( X \). Prove that \( |X| \geqslant C_{n}^{k-1} \). Additionally, when equality holds, \( r = C_{n-1}^{k-1} \).

Answer: \binom{n-1}{k-1}


Compute the limit of the function:

$$
\lim _{x \rightarrow 1}\left(\ln ^{2} e x\right)^{\frac{1}{x^{2}+1}}
$$

Answer: 1


From the $8$ vertices of a cube, choose any $4$ vertices, the probability that these $4$ points lie in the same plane is ______.

Answer: \frac{6}{35}


In our province, the new college entrance examination adopts a "choose 3 out of 7" model, which means choosing 3 subjects from politics, history, geography, physics, chemistry, biology, and technology as elective subjects. How many possible combinations of elective subjects are there? If person A must choose physics and politics, and person B does not choose technology, how many combinations are there such that both persons have at least one subject in common? (Answer mathematically)

Answer: 92


Let \( a, b, c \) be non-negative real numbers. What are the minimum and maximum values of the algebraic expression
\[
s = \sqrt{\frac{ab}{(b+c)(c+a)}} + \sqrt{\frac{bc}{(c+a)(a+b)}} + \sqrt{\frac{ca}{(a+b)(b+c)}}?
\]

Answer: \frac{3}{2}


Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.).

Answer: 205_{16}


Given triangle \( ABC \). On the side \( AC \), which is the largest in the triangle, points \( M \) and \( N \) are marked such that \( AM = AB \) and \( CN = CB \). It turns out that angle \( NBM \) is three times smaller than angle \( ABC \). Find \( \angle ABC \).

Answer: 108


Find the radius of a circle in a segment of which, corresponding to a chord of length $6 \textrm{ cm}$, a square with a side of $2 \textrm{ cm}$ is inscribed.

Answer: \sqrt{10}


There are several weights, each of which has an integer mass. It is known that they can be divided into $k$ groups of equal mass.

Prove that there are at least $k$ ways to remove one weight so that the remaining weights cannot be divided into $k$ groups of equal mass.

Answer: 1


Define the sequence \(a_{0}, a_{1}, \ldots\) as follows:

\[
a_{0} = a_{1} = 1, \quad a_{n+1} = 2 a_{n} + a_{n-1} \quad (n = 1, 2, \ldots)
\]

Prove that \(2\left(a_{2n}^{2} - 1\right)\) is a perfect square for all positive integers \(n\).

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


Given a cyclic quadrilateral \(ABCD\) with a circumradius of \(200\sqrt{2}\) and sides \(AB = BC = CD = 200\). Find the length of side \(AD\).

Answer: 500


The graph of \( r = 2 + \cos 2 \theta \) and its reflection over the line \( y = x \) bound five regions in the plane. Find the area of the region containing the origin.

Answer: \frac{9\pi}{2}-8


On the diagonal \( BD \) of parallelogram \( ABCD \), points \( P \) and \( Q \) are chosen such that \( BP = PQ = QD \). Lines \( CP \) and \( CQ \) intersect sides \( AB \) and \( AD \) at points \( M \) and \( N \) respectively. Find the ratio \( MN : BD \).

Answer: 1:2


Let \( m \) and \( n \) be positive integers such that \( 2001 m^{2} + m = 2002 n^{2} + n \). Prove that \( m - n \) is a perfect square.

Answer: -n


Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$.

Answer: -2


A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. A set of plates in which each possible sequence appears exactly once contains N license plates. Find $\frac{N}{10}$.

Answer: 372


For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$?

Answer: 10


The vertex \( A \) of the parallelogram \( ABCD \) is connected by segments to the midpoints of sides \( BC \) and \( CD \). One of these segments turned out to be twice as long as the other. Determine the nature of the angle \( BAD \): acute, right, or obtuse.

Answer: obtuse


Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter?

Answer: 120


Suppose that the angles of triangle $PQR$ satisfy
\[\cos 3P + \cos 3Q + \cos 3R = 1.\]Two sides of the triangle have lengths 12 and 15.  Find the maximum length of the third side.

Answer: 27


Given an integer  $n\ge 2$ , compute  $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$ , where all  $n$ -element permutations are considered, and where  $\ell(\sigma)$  is the number of disjoint cycles in the standard decomposition of  $\sigma$ .

Answer: n!


Let \(\alpha, \beta, \gamma\) be positive real numbers such that \(\alpha + \beta + \gamma < \pi\), \(\alpha + \beta > \gamma\), \(\beta + \gamma > \alpha\), \(\gamma + \alpha > \beta\). Prove that with the segments of lengths \(\sin \alpha, \sin \beta, \sin \gamma\) we can construct a triangle and that its area is not greater than
$$
\frac{1}{8}(\sin 2\alpha + \sin 2\beta + \sin 2\gamma).
$$

Answer: \frac{1}{8}(\sin2\alpha+\sin2\beta+\sin2\gamma)


Let \( ABC \) be a triangle with \( AB=4 \), \( BC=8 \), and \( CA=5 \). Let \( M \) be the midpoint of \( BC \), and let \( D \) be the point on the circumcircle of \( ABC \) such that segment \( AD \) intersects the interior of \( ABC \), and \( \angle BAD = \angle CAM \). Let \( AD \) intersect side \( BC \) at \( X \). Compute the ratio \( \frac{AX}{AD} \).

Answer: \frac{9}{41}


For what smallest natural $k$ is the number \( 2016 \cdot 20162016 \cdot 201620162016 \cdot \ldots \cdot 20162016\ldots2016 \) (with $k$ factors) divisible by \(3^{67}\)?

Answer: 34


Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$?

Answer: -100


Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1$ and $\|\mathbf{c}\| = 2.$  Find the maximum value of
\[\|\mathbf{a} - 2 \mathbf{b}\|^2 + \|\mathbf{b} - 2 \mathbf{c}\|^2 + \|\mathbf{c} - 2 \mathbf{a}\|^2.\]

Answer: 42


In a castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?

Answer: 13


A passenger train departs from point $A$ towards point $B$ at a speed of $60 \mathrm{~km/h}$. A high-speed train, which departs later, must catch up with the passenger train at point $B$ and travels at a speed of $120 \mathrm{~km/h}$. After completing $\frac{2}{3}$ of its journey, the passenger train continues at half its original speed. The high-speed train catches up with the passenger train 80 kilometers before point $B$. How far is point $B$ from point $A$?

Answer: 360


Find the maximal possible value of  $n$  such that there exist points  $P_1,P_2,P_3,\ldots,P_n$  in the plane and real numbers  $r_1,r_2,\ldots,r_n$  such that the distance between any two different points  $P_i$  and  $P_j$  is  $r_i+r_j$ .

Answer: 4


Compute the smallest positive integer $k$ such that 49 divides $\binom{2 k}{k}$.

Answer: 25


Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool.

Answer: 40


A square board is divided by a grid of horizontal and vertical lines into $n^{2}$ cells with a side length of 1. For what maximum value of $n$ can you mark $n$ cells so that every rectangle with an area of at least $n$ and sides along the grid lines contains at least one marked cell?

Answer: 7


Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the value of $x$.

Answer: 14


A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$?

Answer: \frac{27}{220}


Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.

Answer: 728


The function $f$ is given by the table
\[\begin{tabular}{|c||c|c|c|c|c|}  \hline   x & 1 & 2 & 3 & 4 & 5 \\   \hline  f(x) & 4 & 1 & 3 & 5 & 2 \\  \hline \end{tabular}\]
If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find $u_{2002}$
$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qquad \text{(D) }4 \qquad \text{(E) }5$

Answer: \textbf{(B)}2


$13$  fractions are corrected by using each of the numbers  $1,2,...,26$  once.**Example:** $\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?

Answer: 12


A circle passing through the vertices $A$, $B$, and $C$ of parallelogram $ABCD$ intersects the lines $AD$ and $CD$ at points $M$ and $N$ respectively. The distances from point $M$ to vertices $B$, $C$, and $D$ are 4, 3, and 2 respectively. Find $MN$.

Answer: \frac{8}{3}


Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), respectively. Let point \( P \) lie on the hyperbola \( C \), and let \( G \) and \( I \) be the centroid and incenter of the triangle \( \triangle F_{1} P F_{2} \), respectively. If \( G \) and \( I \) are parallel to the x-axis, determine the circumradius \( R \) of the triangle \( \triangle F_{1} P F_{2} \).

Answer: 5


In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.

Answer: 7


Calculate the integral

$$
I=\int_{0}^{\infty} \frac{\sin a x}{x\left(x^{2}+b^{2}\right)} d x \\
(a>0, b>0)
$$

Answer: \frac{\pi}{2b^2}(1-e^{-})


Given a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimeter of one of the small triangles to the perimeter of the large rectangle.

Answer: \frac{2}{3}


Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triangle is less than 2004.

Answer: 22


When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance between points $A$ and $B$ in meters?

Answer: 300


Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$\begin{aligned} & 3a+5b \equiv 19 \quad(\bmod n+1) \\ & 4a+2b \equiv 25 \quad(\bmod n+1) \end{aligned}$$ Find $2a+6b$.

Answer: 96


Let  $a$  and  $b$  be positive whole numbers such that  $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$ .
Find the fraction  $\frac{a}{b}$  for which the sum  $a+b$  is as small as possible.
Justify your answer

Answer: 3/7


Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\dots$, $C_6$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Let $n$ be the product of the lengths of these twelve chords. Find the remainder when $n$ is divided by $1000$.

Answer: 672


Several circles are positioned in the plane such that there are exactly 12 points where at least two circles intersect. What is the minimum number of regions into which these circles divide the plane?

Answer: 14


In a herd consisting of horses, two-humped camels, and one-humped camels, there are a total of 200 humps. How many animals are in the herd if the number of horses is equal to the number of two-humped camels?

Answer: 200


If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$?

Answer: 356


A circle with diameter \(AC\) has a tangent \(BC\). The line segment \(AB\) intersects the circle at point \(D\). Another tangent to the circle is drawn through point \(D\), intersecting the line segment \(BC\) at point \(K\). In what ratio does point \(K\) divide the segment \(BC\)?

Answer: 1:1


Let \( t = 2016 \) and \( p = \ln 2 \). Evaluate in closed form the sum
\[
\sum_{k=1}^{\infty}\left(1 - \sum_{n=0}^{k-1} \frac{e^{-t} t^n}{n!}\right)(1-p)^{k-1} p.
\]

Answer: 1-(\frac{1}{2})^{2016}


For a given positive integer  $m$ , the series   $$ \sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)} $$  evaluates to  $\frac{a}{bm^2}$ , where  $a$  and  $b$  are positive integers. Compute  $a+b$ .

Answer: 7


Given that the area of quadrilateral \(ABCD\) is 9 and point \(P\) is located inside the quadrilateral. Let the centroids of triangles \(\triangle ABP\), \(\triangle BCP\), \(\triangle CDP\), and \(\triangle DAP\) be \(G_1\), \(G_2\), \(G_3\), and \(G_4\) respectively. Find the area of quadrilateral \(G_1G_2G_3G_4\).

Answer: 2


Misha and Masha had the same multi-digit whole number written in their notebooks, ending in 9876. Masha placed a plus sign between the third and fourth digits, counting from the right, while Misha did the same between the fourth and fifth digits, also counting from the right. To the students' surprise, both resulting sums turned out to be the same. What was the original number the students had written down? Provide all possible answers and prove that there are no others.

Answer: 9999876


In triangle \(ABC\), point \(M\) is the midpoint of side \(AC\), and point \(P\) lies on side \(BC\). Segment \(AP\) intersects \(BM\) at point \(O\). It is given that \(BO = BP\).

Find the ratio \(OM : PC\).

Answer: 1:2


A deck of cards contains 52 cards, where each of the suits "diamonds", "clubs", "hearts", and "spades" has 13 cards, marked $2,3, \cdots, 10, J, Q, K, A$. Any two cards of the same suit and adjacent ranks are called "straight flush cards," and $A$ with 2 is also considered a consecutive pair (i.e., $A$ can be used as 1). Determine the number of ways to draw 13 cards from this deck so that each rank appears exactly once and no "straight flush cards" are included.

Answer: 3^{13}-3


Mobile payment (Alipay and WeChat Pay) has gradually become a payment method for people's shopping consumption. To investigate the age structure of citizens using mobile payment, a random survey of 100 citizens was conducted, and a $2\times 2$ contingency table was obtained as follows:

|                  | 35 years old and below (including 35 years old) | Above 35 years old | Total |
|------------------|--------------------------------------------------|--------------------|-------|
| Using mobile payment | 40                                               |                    | 50    |
| Not using mobile payment |                                                  | 40                 |       |
| Total            |                                                  |                    | 100   |

$(1)$ Complete the above $2\times 2$ contingency table and by calculation, determine whether there is a $99.9\%$ certainty that the payment method is related to age.
$(2)$ In the group of people using mobile payment, $10$ people were further sampled using stratified sampling. From these $10$ people, $2$ people were randomly selected. Let $X$ be the number of people under the age of 35 (including 35 years old) among these $2$ people. Find the distribution of $X$ and its expectation.
Reference formula: $K^{2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$, where $n=a+b+c+d$.
Reference critical value table:

| $P(K^{2}\geqslant k)$ | 0.5  | 0.4  | 0.25 | 0.15 | 0.10 | 0.05 | 0.025 | 0.010 | 0.005 | 0.001 |
|-----------------------|------|------|------|------|------|------|-------|-------|-------|-------|
| $k$                   | 0.455| 0.708| 1.323| 2.072| 2.706| 3.841| 5.024 | 6.635 | 7.879 | 10.828|

Answer: \frac{8}{5}


In the ﬁgure on the right,  $O$  is the center of the circle,  $OK$  and  $OA$  are perpendicular to one another,  $M$  is the midpoint of  $OK$ ,  $BN$   is parallel to  $OK$ , and  $\angle AMN=\angle NMO$ . Determine the measure of  $\angle A B N$  in degrees.

[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label(" $A$ ", A, dir(O--A));
label(" $K$ ", K, dir(O--K));
label(" $B$ ", B, dir(O--B));
label(" $N$ ", N, E);
label(" $M$ ", M, S);
label(" $O$ ", O, SE);[/asy]

Answer: 36


The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$.  For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April.  Approximately how many millions of dollars were spent during the summer months of June, July, and August?

Answer: 2.5


Natural numbers from 1 to \( n \) are arranged in a row in a random order. An arrangement is called bad if it is possible to find 10 numbers (not necessarily consecutive) in descending order. All other arrangements are called good. Prove that the number of good arrangements does not exceed \( 81^n \).

Answer: 81^n


Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5,6,7\}$. Find the remainder when $N$ is divided by $1000$.

Answer: 760


There is a four-digit number \( A \). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \( A \), and the smallest possible number is 594 less than \( A \). What is \( A \)?

Answer: 1963


Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$  Find $n \times s.$

Answer: 153


What is the remainder when \(2,468,135,790\) is divided by \(99\)?

Answer: 54


Two three-letter strings,  $aaa$  and  $bbb$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an  $a$  when it should have been a  $b$ , or as a  $b$  when it should be an  $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let  $S_a$  be the three-letter string received when  $aaa$  is transmitted and let  $S_b$  be the three-letter string received when  $bbb$  is transmitted. Let  $p$  be the probability that  $S_a$  comes before  $S_b$  in alphabetical order. When  $p$  is written as a fraction in lowest terms, what is its numerator?

Answer: 532


A given polynomial  $P(t) = t^3 + at^2 + bt + c$  has  $3$  distinct real roots.
If the equation  $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$  has no real roots,
prove that  $P(2013) >\frac{1}{64}$

Answer: P(2013)>\frac{1}{64}


Most countries in the world use the Celsius temperature scale to forecast the weather, but countries like the United States and the United Kingdom still use the Fahrenheit temperature scale. A student looked up data and found the following information in the table:

| Celsius Temperature $x/^{\circ}\mathrm{C}$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ |
|-------------------------------------------|-----|------|------|------|------|------|
| Fahrenheit Temperature $y/^{\circ}F$       | $32$| $50$ | $68$ | $86$ | $104$| $122$|

$(1)$ Analyze whether the correspondence between the two temperature scales is a linear function. Fill in the blank with "yes" or "no".
$(2)$ Based on the data, calculate the Celsius temperature at $0^{\circ}F$.

Answer: -\frac{160}{9}


In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the smallest possible index of this term?

Answer: 3


The incircle touches the sides $AB$ and $AC$ of triangle $ABC$ at points $X$ and $Y$ respectively. Point $K$ is the midpoint of the arc $AB$ of the circumcircle of triangle $ABC$ (not containing point $C$). It turns out that the line $XY$ bisects segment $AK$. What could be the measure of angle $BAC$?

Answer: 120


Santa Claus has 36 identical gifts divided into 8 bags. The number of gifts in each of the 8 bags is at least 1 and all are different. You need to select some of these bags to evenly distribute all their gifts to 8 children, such that all gifts are distributed completely (each child receives at least one gift). How many different selections are there?

Answer: 31


Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.

Answer: 86


In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\).

Answer: 80


A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.

Answer: 2/3


Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\frac{\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\frac{\sqrt{3}}{2}$. Further, the line $y=kx+m$ ($k \neq 0$, $m \neq 0$) intersects the ellipse at two distinct points C and D, and points C and D both lie on the same circle centered at A.
(1) Find the equation of the ellipse;
(2) When $k = \frac{\sqrt{6}}{3}$, find the value of $m$ and the area of triangle $\triangle ACD$.

Answer: \frac{5}{4}


In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.

[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W);
[/asy]

Answer: 7.5


Answer: 1097393685


In  $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is

Answer: 4K


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


The inequality $y-x<\sqrt{x^2}$ is satisfied if and only if
$\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad \textbf{(C) }y^2<2xy\qquad \textbf{(D) }y<0\qquad \textbf{(E) }x>0\text{ and }y<2x$

Answer: \textbf{(A)}y<0


In the tetrahedron \( KLNM \), it is known that \( KL=MN=4 \), \( KM=LN=5 \), and \( KN=ML=6 \). The points \( P, Q, R, S \) are the centers of the circles inscribed in the triangles \( KLM \), \( KLN \), \( KMN \), and \( LMN \). Find the volume of the tetrahedron \( PQRS \). If necessary, round your answer to hundredths.

Answer: 0.29


On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has an area of $\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\left| z + \frac{1}{z} \right|^2.$

Answer: \frac{36}{13}


Let the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \)) satisfy the following conditions:
1. For all \( x \in \mathbb{R} \), \( f(x-4) = f(2-x) \);
2. For \( x \in (0,2) \), \( f(x) \leq \left( \frac{x+1}{2} \right)^2 \);
3. The minimum value of \( f(x) \) over \( \mathbb{R} \) is 0.

Find the maximum value of \( m \) (with \( m > 1 \)) such that there exists \( t \in \mathbb{R} \) where for all \( x \in [1, m] \), \( f(x+t) \leq x \).

Answer: 9


Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.

Answer: 53


A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a factor of 3 rather than 4. If this process is repeated indefinitely, calculate the total area of the shaded triangles.

Answer: 16


The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtract the number, having the reverse order of digits, from this three-digit number, their difference will be a positive number, in which the hundreds digit coincides with the tens digit of the given number. Find this number.

Answer: 421


A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Calculate the volume of this smaller tetrahedron.

Answer: \frac{\sqrt{2}}{12}


Let \( n \) be a given positive integer. Consider the sum
\[ S = \sum_{1 \leqslant i < j \leqslant n} \left|x_i - x_j \right| \quad (0 < x_i < 1, \, i = 1, 2, \ldots, n). \]
Find the maximum possible value of this sum.
(6th Canadian Mathematical Olympiad)

Answer: \lfloor\frac{n^2}{4}\rfloor


In the 100-digit number \(12345678901234567890 \ldots 1234567890\), all digits in odd positions were crossed out. In the resulting 50-digit number, digits in odd positions were crossed out again. This process continued until there were no digits left. Which digit was crossed out last?

Answer: 4


The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$?

Answer: \frac{8k-168}{k-12}


Two circles of radii 4 and 5 are externally tangent to each other and are circumscribed by a third circle. Find the area of the shaded region created in this way. Express your answer in terms of $\pi$.

Answer: 40\pi


Let \( x \) and \( y \) be acute angles such that \( \sin y = 2005 \cos (x + y) \sin x \). Find the greatest possible value of \( \tan y \).

Answer: \frac{2005\sqrt{2006}}{4012}


Nine digits: $1, 2, 3, \ldots, 9$ are written in some order (forming a nine-digit number). Consider all triplets of digits that come consecutively, and find the sum of the corresponding seven three-digit numbers. What is the maximum possible value of this sum?

Answer: 4648


Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest possible values of the mean of \(m_{1}, m_{2}, \ldots, m_{10}\).

Answer: 4.5


In the tetrahedron \(ABCD\), \(AB\perp BC\), \(CD \perp BC\), \(BC=2\), and the angle between the skew lines \(AB\) and \(CD\) is \(60^\circ\). If the circumradius of the tetrahedron \(ABCD\) is \(\sqrt{5}\), then find the maximum volume of this tetrahedron.

Answer: 2\sqrt{3}


In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively.  Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-sqrt(2),2+sqrt(2)); G=(-sqrt(2),sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--E);
pair M=(B+C)/2; pair N=(F+G)/2;
draw(M--N);

label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,E);

label("$E$",E,S); label("$F$",F,S); label("$G$",G,W); label("$H$",H,W);
label("$M$",M,NE); label("$N$",N,SW);

label("$O$",(1,2.4),E);

[/asy]

Answer: \frac{3}{5}


Define a sequence  $(a_n)$  recursively by  $a_1=0, a_2=2, a_3=3$  and  $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$  for  $n \ge 4$ . Determine the prime factorization of  $a_{19702020}$ .

Answer: 3^{6567340}


For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$.
For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$ .
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Answer: \textbf{(D)}\28


The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the $17$ top-rated roller coasters in the world. In the stem-and-leaf plot, $2 \ 20$ represents $2$ minutes, $20$ seconds, which is the same as $140$ seconds. What is the median of this data set? Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&28&28&50&&\\
1&00&02&&&\\
2&20&25&35&43&45\\
3&00&00&00&30&36\\
4&00&00&&&\\
\end{tabular}

Answer: 163


Answer: -1


We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?

Answer: 20


A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.

Answer: 32


Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?

Answer: (-2,0)


In Ms. Johnson's class, each student averages two days absent out of thirty school days. What is the probability that out of any three students chosen at random, exactly two students will be absent and one will be present on a Monday, given that on Mondays the absence rate increases by 10%? Express your answer as a percent rounded to the nearest tenth.

Answer: 1.5\%


Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\in\mathbb{N}^*$), and that $\{a_{2n-1}\}$ is an increasing sequence, $\{a_{2n}\}$ is a decreasing sequence, find the limit $$\lim_{n\to\infty} \frac{a_{2n-1}}{a_{2n}} = \_\_\_\_\_\_.$$

Answer: -\frac{1}{2}


How many subsets $S$ of the set $\{1,2, \ldots, 10\}$ satisfy the property that, for all $i \in[1,9]$, either $i$ or $i+1$ (or both) is in $S$?

Answer: 144


Find the smallest real number \(\lambda\) such that there exists a sequence \(\{a_n\}\) with all terms greater than 1, and for any positive integer \(n\), the following inequality is satisfied:
\[
\prod_{i=1}^{n+1} a_i < a_n^\lambda.
\]

Answer: 4


Find the third side of an acute triangle if its two sides are \( a \) and \( b \) and it is known that the medians to these sides intersect at a right angle.

Answer: \sqrt{\frac{^2+b^2}{5}}


How many essentially different proportions can be made from the four letters $x, y, z, t$?

Answer: 3


We placed $n$ coins with radius $r$ on a round table with radius $R$ such that each coin lies flat on the table. No more coins can be placed on the table. Show that $\frac{1}{2}\left(\frac{R}{r}-1\right) \leq \sqrt{n} \leq \frac{R}{r}$.

Answer: \frac{1}{2}(\frac{R}{r}-1)\le\sqrt{n}\le\frac{R}{r}


A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$

Answer: \frac{85}{7}


The base of the pyramid \( SABC \) is a triangle \( ABC \) such that \( AB = AC = 10 \) cm and \( BC = 12 \) cm. The face \( SBC \) is perpendicular to the base and \( SB = SC \). Calculate the radius of the sphere inscribed in the pyramid if the height of the pyramid is 1.4 cm.

Answer: 12/19


It is known that the numbers \(x, y, z\) form an arithmetic progression in the given order with a common difference \(\alpha = \arccos \frac{5}{9}\), and the numbers \(1 + \cos x, 1 + \cos y, 1 + \cos z\) form a non-constant geometric progression in the given order. Find \(\cos y\).

Answer: -\frac{7}{9}


In the regular tetrahedron \( P-ABC \), points \( D \), \( E \), and \( F \) lie on edges \( PA \), \( PB \), and \( PC \) respectively. If \( PE \neq PF \), and \( DE = DF = \sqrt{7} \), \( EF = 2 \), find the volume of the tetrahedron \( P-DEF \).

Answer: \frac{\sqrt{17}}{8}


The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?

Answer: 8


Given that \( m \) and \( n \) are integers, the point \( P(m, n^2) \) is on the graph of the function
\[
f(x) = (x^2 - 4)(x^2 + 12x + 32) + 4.
\]
Find the number of points \( P \) that satisfy these conditions.

Answer: 6


Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$.

Answer: 752


For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \).

Answer: 13/8


Given that $A, B, C$ are interior angles of a triangle, what is the minimum value of $\cos A (\sin B + \sin C)$?

Answer: -\frac{2}{9}\sqrt{6}


In tetrahedron \(ABCD\), \(\triangle ABD\) is an equilateral triangle, \(\angle BCD = 90^\circ\), \(BC = CD = 1\), \(AC = \sqrt{3}\). Points \(E\) and \(F\) are the midpoints of \(BD\) and \(AC\) respectively. Determine the cosine of the angle between lines \(AE\) and \(BF\).

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


Let  $A$  be a set of numbers chosen from  $1,2,..., 2015$  with the property that any two distinct numbers, say  $x$  and  $y$ , in  $A$  determine a unique isosceles triangle (which is non equilateral) whose sides are of length  $x$  or  $y$ . What is the largest possible size of  $A$ ?

Answer: 10


What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$?

Answer: 4


Find the smallest positive number from the numbers below.
$\textbf{(A)} \ 10-3\sqrt{11} \qquad  \textbf{(B)} \ 3\sqrt{11}-10 \qquad  \textbf{(C)}\ 18-5\sqrt{13}\qquad \textbf{(D)}\ 51-10\sqrt{26}\qquad \textbf{(E)}\ 10\sqrt{26}-51$

Answer: \textbf{(D)}\51-10\sqrt{26}


Let \( S = \{1, 2, \cdots, 2002\} \). For any \( a, b \in S \) (where \( a \) and \( b \) can be the same), it always holds that \( ab \notin S \). Find the maximum value of \( |S| \).

Answer: 1958


A bus leaves the station at exactly 7:43 a.m. and arrives at its destination at exactly 8:22 a.m. on the same day. How long, in minutes, was the bus trip?

Answer: 39\,


Let \( S = \{1, 2, 3, \cdots, 280\} \). Find the smallest positive integer \( n \) such that every subset of \( S \) with \( n \) elements contains 5 pairwise coprime numbers.

Answer: 217


Point $O$ is the center of the circumcircle of triangle $ABC$. Points $Q$ and $R$ are chosen on sides $AB$ and $BC$ respectively. Line $QR$ intersects the circumcircle of triangle $ABR$ a second time at point $P$ and intersects the circumcircle of triangle $BCQ$ a second time at point $S$. Lines $AP$ and $CS$ intersect at point $K$. Find the angle between lines $KO$ and $QR$.

Answer: 90


In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total shaded area in square inches.

Answer: 1296 - 324\pi


$1989$  equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.

Answer: 4


For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]

Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).

Answer: 3


Given that Lucas's odometer showed 27372 miles, which is a palindrome, and 3 hours later it showed another palindrome, calculate Lucas's average speed, in miles per hour, during this 3-hour period.

Answer: 33.33


Petya and Vasya are playing a game. First, Petya thinks of a polynomial \( P(x) \) with integer coefficients. Then, several moves are made. On a move, Vasya pays Petya one ruble and names any integer \( a \) of his choice that he has not previously named, and Petya responds with the number of integer solutions to the equation \( P(x) = a \). Vasya wins as soon as Petya gives the same number twice (not necessarily consecutively). What is the minimum number of rubles Vasya needs to guarantee a win?

Answer: 4


From the digits 1 and 2, five $n$-digit numbers were created such that for each pair of numbers, the digits match in exactly $m$ positions, but in no single position did all five numbers have the same digit. Prove that the ratio $m / n$ is no less than $2 / 5$ and no greater than $3 / 5$.

Answer: \frac{2}{5}\le\frac{}{n}\le\frac{3}{5}


Find the area bounded by the graph of  $ y\equal{}\sqrt[3]{x\plus{}\sqrt{x^2\plus{}1}}\plus{}\sqrt[3]{x\minus{}\sqrt{x^2\plus{}1}}$ , the line  $ x\minus{}y\minus{}1\equal{}0$  and the  $ x$  axis.

Answer: 5/8


Suppose \( f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0} \) and \( g(x) = b_{m} x^{m} + \cdots + b_{1} x + b_{0} \) are two polynomials with integer coefficients. If all coefficients of the product \( f(x) g(x) \) are even but not all divisible by 4, prove that one of the polynomials \( f(x) \) or \( g(x) \) has all even coefficients, and the other has at least one odd coefficient.

Answer: 1


Given is a chessboard 8x8. We have to place  $n$  black queens and  $n$  white queens, so that no two queens attack. Find the maximal possible  $n$ .
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)

Answer: 8


Three people jointly start a business with a total investment of 143 million yuan. The ratio of the highest investment to the lowest investment is 5:3. What is the maximum and minimum amount the third person could invest in millions of yuan?

Answer: 39


Cyclic pentagon  $ ABCDE$  has a right angle  $ \angle{ABC} \equal{} 90^{\circ}$  and side lengths  $ AB \equal{} 15$  and  $ BC \equal{} 20$ . Supposing that  $ AB \equal{} DE \equal{} EA$ , find  $ CD$ .

Answer: 7


The sum of eight numbers equals \( \frac{4}{3} \). It turns out that the sum of any seven out of these eight numbers is positive. What is the smallest integer value that the smallest of these numbers can take?

Answer: -7


For a positive integer \( k \), let \( g(k) \) represent the largest odd factor of \( k \) (for example, \( g(3)=3 \) and \( g(20)=5 \)). Find the sum \( g(1) + g(2) + g(3) + \cdots + g(2^n) \) where \( n \) is a positive integer.

Answer: \frac{4^n+2}{3}


Prove the equation:

\[
\frac{\left(\sin ^{2} \alpha+\tan^{2} \alpha+1\right)\left(\cos ^{2} \alpha-\cot^{2} \alpha+1\right)}{\left(\cos ^{2} \alpha+\cot^{2} \alpha+1\right)\left(\sin ^{2} \alpha+\tan^{2} \alpha-1\right)}=1
\]

Answer: 1


$(HUN 6)$  Find the positions of three points  $A,B,C$  on the boundary of a unit cube such that  $min\{AB,AC,BC\}$  is the greatest possible.

Answer: \sqrt{2}


What is the maximum number of checkers that can be placed on a $6 \times 6$ board so that no three checkers (more precisely, the centers of the cells they occupy) are collinear (in any direction)?

Answer: 12


How many distinct sets of  $5$  distinct positive integers  $A$  satisfy the property that for any positive integer  $x\le 29$ , a subset of  $A$  sums to  $x$ ?

Answer: 4


Given triangle \( \triangle ABC \) with its incenter \( I \). Circles \( \odot O_1 \), \( \odot O_2 \), and \( \odot O_3 \) pass through \( B, C \); \( A, C \); and \( A, B \) respectively, and each intersect \( \odot I \) orthogonally. Circle \( \odot O_1 \) intersects \( \odot O_2 \) at another point \( C' \). Similarly, points \( B' \) and \( A' \) are defined. Prove that the circumradius of \( \triangle A'B'C' \) is equal to half the radius of \( \odot I \).

Answer: \frac{r}{2}


Let  $a_1, a_2, \cdots, a_{2022}$  be nonnegative real numbers such that  $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs  $(i, j)$ ,  $1\leq i,j\leq 2022$ , satisfying   $$ a_i^2+a_j\ge \frac 1{2021}. $$

Answer: 2021\times2022


If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______.

Answer: 14


Given 9 points in space, no four of which are coplanar, find the smallest natural number \( n \) such that if any \( n \) line segments connecting these points are colored either red or blue, there will always be a triangle with all sides of the same color.

Answer: 33


Is it possible to cut an arbitrary rhombus into no more than two pieces so that these pieces can form a rectangle?

Answer: Yes


In each of the following 6-digit positive integers: 555555, 555333, 818811, 300388, every digit in the number appears at least twice. Find the number of such 6-digit positive integers.

Answer: 11754


In triangle \( \triangle ABC \), it is given that \( \angle ABC = 44^\circ \), \( D \) is a point on side \( BC \) such that \( DC = 2 AB \), and \( \angle BAD = 24^\circ \). Find the measure of \( \angle ACB \).

Answer: 22


Let \( n \) be a positive integer and let \( A_{1}, A_{2}, \ldots, A_{2n+1} \) be subsets of set \( B \). Assume that:

a) each \( A_{i} \) has exactly \( 2n \) elements,

b) the intersection \( A_{i} \cap A_{j} \) (for \( 1 \leq i < j \leq 2n+1 \)) has exactly one element,

c) each element of \( B \) is in at least two of the \( A_{i} \) sets.

For which values of \( n \) is it possible to assign each element of \( B \) either the number 0 or 1 so that each \( A_{i} \) contains exactly \( n \) elements assigned the number 0?

(Czechoslovakia)

Answer: n


Each positive integer number  $n \ ge 1$  is assigned the number  $p_n$  which is the product of all its non-zero digits. For example,  $p_6 = 6$ ,  $p_ {32} = 6$ ,  $p_ {203} = 6$ . Let  $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides  $S $ .

Answer: 103


Answer: 4094


It is known that the numbers \( x, y, z \) form an arithmetic progression with the common difference \( \alpha=\arccos \frac{2}{3} \), and the numbers \( \frac{1}{\sin x}, \frac{6}{\sin y}, \frac{1}{\sin z} \) also form an arithmetic progression in the given order. Find \( \sin ^{2} y \).

Answer: \frac{5}{8}


The sum of the non-negative numbers \(x_{1}, x_{2}, \ldots, x_{10}\) is 1. Find the greatest possible value of the sum \(x_{1} x_{2} + x_{2} x_{3} + \ldots + x_{9} x_{10}\).

Answer: 0.25


In  $ \vartriangle ABC$ ,  $\angle B = 46^o$  and  $\angle C = 48^o$  . A circle is inscribed in   $ \vartriangle ABC$  and the points of tangency are connected to form  $PQR$ . What is the measure of the largest angle in  $\vartriangle P QR$ ?

Answer: 67


What is the value of the sum
\[
  \sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

Answer: \frac{49}{4}


Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.

Answer: 215.7


The sides \(AD\) and \(DC\) of the inscribed quadrilateral \(ABCD\) are equal. On the side \(BC\), point \(X\) is marked such that \(AB = BX\). It is known that \(\angle B = 34^\circ\) and \(\angle XDC = 52^\circ\).

(a) (1 point) How many degrees is the angle \(AXC\)?

(b) (3 points) How many degrees is the angle \(ACB\)?

Answer: 47


Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$              .

Answer: -7


Suppose in a right triangle LMN, where angle M is the right angle, $\cos N = \frac{4}{5}$ and length LM is given by 20 units. What is the length of LN?

Answer: 25


Billie has a die with the numbers 1, 2, 3, 4, 5, and 6 on its six faces. Niles has a die which has the numbers 4, 4, 4, 5, 5, and 5 on its six faces. When Billie and Niles roll their dice, the one with the larger number wins. If the two numbers are equal, it is a draw. The probability that Niles wins, when written as a fraction in its lowest terms, is \(\frac{p}{q}\). What is the value of \(7p + 11q\)?

Answer: 181


Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500.

Answer: 1587


Consider a round table on which  $2014$  people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated  $2$  seats away from the person at the head of the table.

Answer: 1/2013


Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.
[asy] pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("$A$",A,dir(0)); label("$B$",B,dir(75)); label("$C$",C,dir(90)); label("$D$",D,dir(105)); label("$E$",E,dir(180)); label("$F$",F,dir(225)); label("$G$",G,dir(260)); label("$H$",H,dir(280)); label("$I$",I,dir(315)); [/asy]

Answer: 58


Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$.
$(1)$ If there are two tangents passing through point $A$, find the range of $m$.
$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilateral $PMCN$.

Answer: \frac{7\sqrt{15}}{5}


Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 \; (a > b > 0)$, one of its vertices is $M(0,1)$, and its eccentricity $e = \dfrac {\sqrt {6}}{3}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Suppose a line $l$ intersects the ellipse at points $A$ and $B$, and the distance from the origin $O$ to line $l$ is $\dfrac {\sqrt {3}}{2}$, find the maximum area of $\triangle AOB$.

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


Let positive integers \( a, b, c, d \) satisfy \( a > b > c > d \) and \( a+b+c+d=2004 \), \( a^2 - b^2 + c^2 - d^2 = 2004 \). Find the minimum value of \( a \).

Answer: 503


Given that $\{a_n\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \cdot a_{24} < 0$, determine the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$.

Answer: 46


Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$.

Answer: 20


Point  $P$  and equilateral triangle  $ABC$  satisfy  $|AP|=2$ ,  $|BP|=3$ . Maximize  $|CP|$ .

Answer: 5


On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.
Which city had the greatest percentage increase in population from $1970$ to $1980$?

Answer: C


In a football tournament, there were 15 teams, and each team played with every other team exactly once. A victory awarded 3 points, a draw awarded 1 point, and a loss awarded 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least $N$ points. What is the maximum integer value that $N$ can take?

Answer: 34


There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers?

Answer: 10890


Let  $S$  be the set of sequences of length 2018 whose terms are in the set  $\{1, 2, 3, 4, 5, 6, 10\}$  and sum to 3860. Prove that the cardinality of  $S$  is at most
\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]

Answer: 2^{3860}\cdot(\frac{2018}{2048})^{2018}


The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay?

Answer: 2800


In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament?

Answer: 15


John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green?

Answer: 110


Let \(ABC\) be a triangle with \(AB = AC\). The angle bisectors of \(A\) and \(B\) meet the sides \(BC\) and \(AC\) in \(D\) and \(E\), respectively. Let \(K\) be the incenter of triangle \(ADC\). Suppose that \(\angle BEK = 45^\circ\). Find all possible values of \(\angle BAC\).

Answer: 90


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $(2a-c)\cos B = b\cos C$;
(1) Find the magnitude of angle $B$;
(2) Let $\overrightarrow{m}=(\sin A, \cos 2A), \overrightarrow{n}=(4k,1)$ ($k>1$), and the maximum value of $\overrightarrow{m} \cdot \overrightarrow{n}$ is 5, find the value of $k$.

Answer: \frac{3}{2}


Find all positive roots of the equation \( x^{x} + x^{1-x} = x + 1 \).

Answer: 1


In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The vectors $m=(\cos (A-B),\sin (A-B))$, $n=(\cos B,-\sin B)$, and $m\cdot n=-\frac{3}{5}$.

(1) Find the value of $\sin A$.
(2) If $a=4\sqrt{2}$, $b=5$, find the measure of angle $B$ and the projection of vector $\overrightarrow{BA}$ onto the direction of $\overrightarrow{BC}$.

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


There are positive integers  $m$  and  $n$  such that  $m^2 -n = 32$  and   $\sqrt[5]{m +\sqrt{n}}+  \sqrt[5]{m -\sqrt{n}}$  is a real root of the polynomial  $x^5 - 10x^3 + 20x - 40$ . Find  $m + n$ .

Answer: 388


(1) Evaluate the expression $$(\frac{\sqrt{121}}{2018} - 5)^0 + 2^{-2} \cdot (2\frac{1}{4})^{-\frac{1}{2}} - \log_4 3 \cdot \log_3 \sqrt{8}$$;
(2) The function $f(x) = x^{2-m}$ is an odd function defined on $[-3-m, m^2-m]$. Find the value of $f(m)$.

Answer: -1


Find the integer that is closest to $2000\sum_{n=2}^{5000}\frac{1}{n^2-1}$.

Answer: 1500


In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $2a\cos A=c\cos B+b\cos C$.

1. Find the value of $\cos A$.
2. If $a=1$ and $\cos^{2}\frac{B}{2}+\cos^{2}\frac{C}{2}=1+\frac{\sqrt{3}}{4}$, find the length of side $c$.

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


Let \( N \) be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to \( N \), independently and uniformly at random. Let \( p_N \) denote the probability that the product of these two integers has a units digit of 0. The maximum possible value of \( p_N \) over all possible choices of \( N \) can be written as \( \frac{a}{b} \), where \( a \) and \( b \) are relatively prime positive integers. Compute \( 100a + b \).

Answer: 2800


$x, y, z$  are positive reals such that  $x \leq 1$ . Prove that  $$ xy+y+2z \geq 4 \sqrt{xyz} $$

Answer: xy+y+2z\ge4\sqrt{xyz}


\(PQRS\) is a square. The points \(T\) and \(U\) are the midpoints of \(QR\) and \(RS\) respectively. The line \(QS\) cuts \(PT\) and \(PU\) at \(W\) and \(V\) respectively. What fraction of the total area of the square \(PQRS\) is the area of the pentagon \(RTWVU\)?

A) \(\frac{1}{3}\)

B) \(\frac{2}{5}\)

C) \(\frac{3}{7}\)

D) \(\frac{5}{12}\)

E) \(\frac{4}{15}\)

Answer: \frac{1}{3}


Let \( P-ABC \) be a regular triangular pyramid with height \( PO \). M is the midpoint of \( PO \). A plane through \( AM \) is parallel to edge \( BC \), dividing the pyramid into two parts. Find the volume ratio of these two parts.

Answer: \frac{4}{21}


Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum possible value of \(A\).

Answer: 4648


Griffin and Hailey run for $45$ minutes on a circular track. Griffin runs counterclockwise at $260 m/min$ and uses the outer lane with a radius of $50$ meters. Hailey runs clockwise at $310 m/min$ and uses the inner lane with a radius of $45$ meters, starting on the same radial line as Griffin. Determine how many times do they pass each other.

Answer: 86


A square with sides 8 inches is shown. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{HD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, H, Q;
A = (0,0); B= (2,0); C = (1,2); D = (2,2); H = (0,2); Q = (1,1);
draw(A--B--D--H--cycle);
draw(C--Q); draw(Q--A); draw(Q--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$Q$",Q,NW);label("$H$",H,NW);
label("$8''$",(1,0),S);

[/asy]

Answer: 12


Suppose that $\sec y - \tan y = \frac{15}{8}$ and that $\csc y - \cot y = \frac{p}{q},$ where $\frac{p}{q}$ is in lowest terms. Find $p+q.$

Answer: 30


Construct the three altitudes of a given triangle with the minimum number of construction steps! Count a step as the drawing of a visible line: a circle, a circular arc, or a straight line; do not count the selection of the compass span, placing the compass point, or appropriately positioning the straightedge (even though these actions require attention, they are not visible).

Answer: 7\,


Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase?

Answer: 33


Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers.

Answer: 104


In an acute-angled triangle \( \triangle ABC \), let \( H \) be the orthocenter and \( O \) be the circumcenter. The perpendicular bisector of \( CH \) intersects \( AC \) and \( BC \) at points \( X \) and \( Y \), respectively. The lines \( XO \) and \( YO \) intersect \( AB \) at points \( P \) and \( Q \), respectively. Given that \( XP + YQ = AB + XY \), find \( \angle OHC \).

Answer: 90


A circle with radius \( r \) is surrounded by a circle with radius \( R \). The area of the larger circle is \(\frac{a}{b}\) times the area of the region between the two circles (\(a > b\)). What is the ratio \( R: r \)?

(A) \(\sqrt{a}: \sqrt{b}\).

(B) \(\sqrt{a}: \sqrt{a-b}\).

(C) \(\sqrt{b}: \sqrt{a-b}\).

(D) \(a: \sqrt{a-b}\).

(E) \(b: \sqrt{a-b}\).

(20th Annual High School Mathematics Examination, 1969)

Answer: \sqrt{}:\sqrt{-b}


Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, with $A$ being the left vertex, and $P$ is a point on the ellipse $E$ such that the circle with diameter $PF_1$ passes through $F_2$ and $|PF_{2}|= \frac {1}{4}|AF_{2}|$, determine the eccentricity of the ellipse $E$.

Answer: \frac{3}{4}


Given a sequence of positive terms $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\{c_n\}$ where $c_n = (-1)^n \frac{2a_n + 1}{2S_n}$. Find the sum of the first 2016 terms of the sequence $\{c_n\}$.

Answer: - \frac{2016}{2017}


For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4?

Answer: 34


One of the students wrote on the board $1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 = 21$, where instead of asterisks he placed "+" or "-". Another student changed several signs to the opposite ones and as a result got 20. Can we assert that at least one of the students made a mistake in calculating the result?

Answer: 1


A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is:
$\text{(A) } 1\quad \text{(B) } 15\quad \text{(C) } 15\tfrac{1}{3}\quad \text{(D) } 15\tfrac{1}{2}\quad \text{(E) } 15\tfrac{3}{4}$

Answer: \textbf{(B)}15


Given the quadratic equation with real coefficients \( x^{2} + ax + 2b - 2 = 0 \), the equation has two distinct real roots, one of which is in the interval \( (0, 1) \) and the other in the interval \( (1, 2) \). Determine the range of values for \( \frac{b-4}{a-1} \).

Answer: (\frac{1}{2},\frac{3}{2})


There are $N = 1998$ cities in a country, and from each city, there are direct flights to three other cities (all flights are bidirectional). It is known that from each city, with a few transfers, one can reach any other city. The Ministry of Security wants to declare 200 cities closed, with no two of them connected by an airline. Prove that this can be done in such a way that it remains possible to travel from any open city to any other open city without having to transfer through the closed cities.

Answer: 200


Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$.

Answer: 3\pi


The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:

1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.

How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45.

Answer: 35


A function $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ is linear in each of the $x_{i}$ and $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{1}{x_{1} x_{2} \cdots x_{n}}$ when $x_{i} \in\{3,4\}$ for all $i$. In terms of $n$, what is $f(5,5, \ldots, 5)$?

Answer: \frac{1}{6^{n}}


What is the smallest $n > 1$ such that there exists a palindromic $n$-string, where an $n$-string is a string of digits formed by writing the numbers $1, 2, \ldots, n$ in some order (in base ten)?

Answer: 19


Let \(P\) be any point on the angle bisector of \(\angle A\) in the triangle \(ABC\), and let \(A', B', C'\) be the respective points on the lines \(BC, CA,\), and \(AB\) such that \(PA'\) is perpendicular to \(BC\), \(PB'\) is perpendicular to \(CA\), and \(PC'\) is perpendicular to \(AB\). Prove that \(PA'\) and \(B'C'\) intersect on the median \(AM\), where \(M\) is the midpoint of \(BC\).

Answer: 0


Find the biggest positive integer  $n$  such that  $n$  is  $167$  times the amount of it's positive divisors.

Answer: 2004


Determine all integers  $n \ge 2$  such that  $n \mid s_n-t_n$  where  $s_n$  is the sum of all the integers in the interval  $[1,n]$  that are mutually prime to  $n$ , and  $t_n$  is the sum of the remaining integers in the same interval.

Answer: n


Given a trapezoid \(ABCD\). A line parallel to the bases intersects the legs \(AB\) and \(CD\) at points \(M\) and \(N\) respectively. A point \(E\) is taken on the base \(BC\). The segments \(AE\) and \(ED\) intersect \(MN\) at points \(S\) and \(T\). The areas of triangles \(AMS\), \(SET\), and \(TND\) are 12, 8, and 15, respectively. What is the minimum possible area of the trapezoid given these conditions?

Answer: 125


If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are:

Answer: 1, 8, and 49