In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$. a) Prove $SX,TY, AD$ are concurrent at a point $Z$. b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$. [i]Ray Li.[/i]

Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.

The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who? [i]Remark.[/i] The answer may depend on initial position of the checker.

A convex quadrilateral $ ABCD$ is inscribed in a circle. The tangents to the circle through $ A$ and $ C$ intersect at a point $ P$, such that this point $ P$ does not lie on $ BD$, and such that $ PA^{2}=PB\cdot PD$. Prove that the line $ BD$ passes through the midpoint of $ AC$.

A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if [list][b](a)[/b] one angle of the triangle is three times as big as another; [b](b)[/b] one angle of the triangle is obtuse and is twice as big as one of the acute angles?[/list]

Let be an odd integer $ n\ge 3 $ and an $ n\times n $ real matrix $ A $ whose determinant is positive and such that $ A+\text{adj} A=2A^{-1} . $ Prove that $ A^{2010} +\text{adj}^{2010} A =2A^{-2010} . $ [i]Lucian Petrescu[/i]

In acute, scalene Triangle $ABC$, $H$ is orthocenter,$ BD$ and $CE$ are heights. $X,Y$ are reflection of $A$ from $D$,$E$ respectively such that the points$ X,Y$ are on segments $DC$ and $EB$. The intersection of circles $ HXY$ and $ADE$ is $F.$ ( $F \neq H$). Prove that$ AF$ intersects middle point of $BC$. ( $M$ in the diagram is Midpoint of $BC$)

Let be a group with $ 10 $ elements for which there exist two non-identity elements, $ a,b, $ having the property that $ a^2 $ and $ b^2 $ are the identity. Show that this group is not commutative.

Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home? [asy]pair J=(0,6), K=origin, L=(10,0), M=(10,6); draw(J--K--L--M--cycle); label("$J$", J, dir((5,3)--J)); label("$K$", K, dir((5,3)--K)); label("$L$", L, dir((5,3)--L)); label("$M$", M, dir((5,3)--M));[/asy] $\textbf{(A)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(15,15)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(B)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw((0,6)--(1,6)--(1,12)--(2,12)--(2,11)--(3,11)--(3,1)--(12,1)--(12,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(C)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2.7,8)--(3,9)^^(11,9)--(14,0)); draw(Arc((4,9), 1, 0, 180)); draw(Arc((10,9), 1, 0, 180)); draw(Arc((7,9), 2, 180,360)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(D)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2,6)--(7,14)--(10,12)--(14,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(E)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(3,6)--(7,6)--(10,12)--(14,12)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy]

For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

Andy chooses not necessarily distinct digits $G$, $U$, $N$, and $A$ such that the $5$ digit number $GUNGA$ is divisible by $44$. Find the least possible value of $G+U+N+G+A$. [i]Proposed by Andy Xu[/i]

Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$ What is the minimal possible $s$? What $x_i $correspond it?

A natural number $n$ is given. For all integer triplets $(a,b,c)$ such that $0 < |a| , |b|, |c| < 2023$ and satisfying below, show that the product of all possible integer $a$ is a perfect square. (The value of $a$ allows duplication) $$(a+nb)(a-nc) + abc = 0$$

(1) Prove that $\sqrt[3]{2}$ is irrational. (2) Let $P(x)$ be a polynomoal with rational coefficients such that $P(\sqrt[3]{2})=0$. Prove that $P(x)$ is divisible by $x^3-2$. 35 points

Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.

Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.

Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

On the side $ \overline{DC} $ of the rectangle $ ABCD $ in which $ \frac{AB}{AD} = \sqrt{2} $ a semicircle is built externally. Any point $ M $ of the semicircle is connected by segments with $ A $ and $ B $ to obtain points $ K $ and $ L $ on $ \overline{DC} $, respectively. Prove that $ DL^2 + KC^2 = AB^2 $.

Compute $\left\lfloor\frac{1}{\frac{1}{2022}+\frac{1}{2023}+\dots+\frac{1}{2064}}\right\rfloor$.

Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively. [list=a] [*]Prove that the points $M, N, P, Q, R$ lie on the same plane. [*]Find the measure of the angle between the planes $(MNP)$ and $(ABC)$.[/list]

There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set $S{}$ of vertices [i]delightful[/i] if no two of its vertices are connected by an edge, but any vertex not from $S{}$ is connected to at least one vertex from $S{}$. For which smallest $m$ is there necessarily a delightful set of at most $m$ vertices?

Let $ABC$ be a triangle and let $O$ be the centre of its circumscribed circle. Points $X, Y$ which are neither of the points $A, B$ or $C$, lie on the circumscribed circle and are so that the angles $XOY$ and $BAC$ are equal (with the same orientation). Show that the orthocentre of the triangle that is formed by the lines $BY, CX$ and $XY$ is a fixed point.

Find all infinite sequences of real numbers $ \left( a_n \right)_{n\ge 1} $ that verify, for any natural number $ n, $ the inequalities $$ \frac{1}{2\sqrt{a_{n+1}}} <\sqrt{n+1} -\sqrt{n} <\frac{1}{ 2\sqrt{a_n}} . $$

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board can be attacked by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less. In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?