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?

There are $n$ players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$ (where $k$ is a fixed integer). Find the sum of all possible values of $n$.

Prove the equality $$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$ (R. Zhenodarov)

Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.

Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively. 1. Prove that triangles $IEF$ and $OKH$ are similar. 2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.

Let $\mathbb{R}[x,y]$ denote the set of two-variable polynomials with real coefficients. We say that the pair $(a,b)$ is a [i]zero[/i] of the polynomial $f \in \mathbb{R}[x,y]$ if $f(a,b)=0$. If polynomials $p,q \in \mathbb{R}[x,y]$ have infinitely many common zeros, does it follow that there exists a non-constant polynomial $r \in \mathbb{R}[x,y]$ which is a factor of both $p$ and $q$?