Given is a triangle $ABC$ with circumcircle $\gamma$. Points $E, F$ lie on $AB, AC$ such that $BE=CF$. Let $(AEF)$ meet $\gamma$ at $D$. The perpendicular from $D$ to $EF$ meets $\gamma$ at $G$ and $AD$ meets $EF$ at $P$. If $PG$ meets $\gamma$ at $J$, prove that $\frac {JE} {JF}=\frac{AE} {AF}$.

Given $ n$ points in a line so that any distance occurs at most twice, show that the number of distance occurring exactly once is at least $ \lfloor n/2 \rfloor$. [i]V. T. Sos, L. Szekely[/i]

In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

Let be eight real numbers $ 1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4. $ Prove that $$ \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\ a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\ a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\ a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\ \end{vmatrix} >0. $$ [i]Marian Andronache, Ion Savu[/i]

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards. Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers. [i]Proposed by ABCDE and Tristan Shin

The sides of a triangle with positive area have lengths $ 4$, $ 6$, and $ x$. The sides of a second triangle with positive area have lengths $ 4$, $ 6$, and $ y$. What is the smallest positive number that is [b]not[/b] a possible value of $ |x \minus{} y|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$

Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.

Provided that $\{a_i\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolute value of $\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.

Near the end of a game of Fish, Celia is playing against a team consisting of Alice and Betsy. Each of the three players holds two cards in their hand, and together they have the Nine, Ten, Jack, Queen, King, and Ace of Spades (this set of cards is known by all three players). Besides the two cards she already has, each of them has no information regarding the other two's hands (In particular, teammates Alice and Betsy do not know each other's cards). It is currently Celia's turn. On a player's turn, the player must ask a player on the other team whether she has a certain card that is in the set of six cards but [i]not[/i] in the asker's hand. If the player being asked does indeed have the card, then she must reveal the card and put it in the asker’s hand, and the asker shall ask again (but may ask a different player on the other team); otherwise, she refuses and it is now her turn. Moreover, a card may not be asked if it is known (to the asker) to be not in the asked person's hand. The game ends when all six cards belong to one team, and the team with all the cards wins. Under optimal play, the probability that Celia wins the game is $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$. [i]Proposed by Yannick Yao[/i]

In a board of $2000\times2001$ squares with integer coordinates $(x,y)$, $0\leq{x}\leq1999$ and $0\leq{y}\leq2000$. A ship in the table moves in the following way: before a move, the ship is in position $(x,y)$ and has a velocity of $(h,v)$ where $x,y,h,v$ are integers. The ship chooses new velocity $(h^\prime,v^\prime)$ such that $h^\prime-h,v^\prime-v\in\{-1,0,1\}$. The new position of the ship will be $(x^\prime,y^\prime)$ where $x^\prime$ is the remainder of the division of $x+h^\prime$ by $2000$ and $y^\prime$ is the remainder of the division of $y+v^\prime$ by $2001$. There are two ships on the board: The Martian ship and the Human trying to capture it. Initially each ship is in a different square and has velocity $(0,0)$. The Human is the first to move; thereafter they continue moving alternatively. Is there a strategy for the Human to capture the Martian, independent of the initial positions and the Martian’s moves? [i]Note[/i]: The Human catches the Martian ship by reaching the same position as the Martian ship after the same move.

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

Misha has a $100$x$100$ chessboard and a bag with $199$ rooks. In one move he can either put one rook from the bag on the lower left cell of the grid, or remove two rooks which are on the same cell, put one of them on the adjacent square which is above it or right to it, and put the other in the bag. Misha wants to place exactly $100$ rooks on the board, which don't beat each other. Will he be able to achieve such arrangement?

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$

There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. In how many ways Can you add three integers Summing seventeen? Order matters here. For example, eight, three, six Is not eight, six, three. All nonnegative, Do not need to be distinct. What is your answer? [i]Proposed by Derek Gao[/i]

Let $n$ be a positive integer, such that $125n+22$ is a power of $3$. Prove that $125n+29$ has a prime factor greater than $100$.

There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions: (i) Every groups contains an equal number of boys and girls. (ii) There is no connected pair in the same group. Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.

At each node of the checkboard $n\times n$ board, a beetle sat. At midnight, each beetle crawled into the center of a cell. It turned out that the distance between any two beetles sitting in the adjacent (along the side) nodes didn't increase. Prove that at least one beetle crawled into the center of a cell at the vertex of which it sat initially. [i](A. Voidelevich)[/i]

In $\triangle ABC$, $AB=10$, $\angle A=30^\circ$, and $\angle C=45^\circ$. Let $H,D$, and $M$ be points on line $\overline{BC}$ such that $\overline{AH}\perp\overline{BC}$, $\angle BAD=\angle CAD$, and $BM=CM$. Point $N$ is the midpoint of segment $\overline{HM}$, and point $P$ is on ray $AD$ such that $\overline{PN}\perp\overline{BC}$. Then $AP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.

Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$ [i]M. Zorka[/i]

A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.

Find all integral solutions of the equation $$|p^r-q^s|=1,$$ where $p$ and $q$ are prime numbers and $r$ and $s$ are positive integers larger than unity. Prove that there are no other solutions.