Two circles intersect at points $P$ and $Q$. The straight line intersects these circles at points $A$, $B$, $C$, $D$, as shown in fig. . Prove that $\angle APB = \angle CQD$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/a581e11be68bbb628db5b5b8e75c7ff6e196c5.png[/img]

Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]

On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?

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

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

Bamicin is initially at $(20, 20)$ in a cartesian plane. Every minute, if Bamicin is at point $P$, Bamicin can move to a lattice point exactly $37$ units from $P$. Determine all lattice points Bamicin can visit.

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$. [i](Proposed by David Anghel, Romania)[/i]

Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that [list=i] [*] $f(m,m)=m$ [*] $f(m,k) = f(k,m)$ [*] $f(m, m+k) = f(m,k)$[/list] , for each $m,k \in \mathbb{Z}^+$.

In triangle $ABC$, $\angle A = 60^o$,$ \angle B = 45^o$. On the sides $AC$ and $BC$ points $M$ and $N$ are taken, respectively, so that the straight line $MN$ cuts off a triangle similar to this one. Find the ratio of $MN$ to $AB$ if it is known that $CM : AM = 2:1$.

Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$? $\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$

Let $ABCDEFGH$ be a regular octagon with side length $\sqrt{60}$. Let $\mathcal{K}$ denote the locus of all points $K$ such that the circumcircles (possibly degenerate) of triangles $HAK$ and $DCK$ are tangent. Find the area of the region that $\mathcal{K}$ encloses.

Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$. Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$. Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value. [i]Proposed by Ivan Chan Kai Chin[/i]

An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club? $ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 $

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

In the following grid below, each row and column contains the numbers $1,2,3,4,5$ exactly once. Furthermore, each of the three sections have the same sum. Find, with proof, all possible ways to fill the grid in. [asy] unitsize(0.5 cm); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0),linewidth(3)); draw((1,0)--(1,1)--(0,1)); draw((2,0)--(2,3)--(0,3)); draw((3,0)--(3,2)--(0,2)); draw((2,5)--(2,4)--(5,4)); draw((3,5)--(3,3)--(5,3)); draw((4,5)--(4,2)--(5,2)); draw((4,0)--(4,1)); draw((1,5)--(1,4)); draw((0,4)--(1,4)--(1,1)--(5,1),linewidth(3)); draw((1,4)--(2,4)--(2,3)--(3,3)--(3,2)--(4,2)--(4,1),linewidth(3)); [/asy]

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by Michael Ren[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Consider an urn containing $51$ white and $50$ black balls. Every turn, we randomly pick a ball, record the color of the ball, and then we put the ball back into the urn. We stop picking when we have recorded $n$ black balls, where $n$ is an integer randomly chosen from $\{1, 2,... , 100\}$. What is the expected number of turns?

Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.

It is known that $\cos 157^o = a$, where $a$ is given. Calculate $1^o$ in terms of $a$.

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?