Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through $H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines $BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$ respectively, lies on the circumcircle of triangle $HXY$. Proposed by [i]Danilo Khilko[/i]

Let $ABCD$ be a square. Let $P$ be a point on the semicircle of diameter $AB$ outside the square. Let $M$ and $N$ be the intersections of $PD$ and $PC$ with $AB$, respectively. Prove that $MN^2 = AM \cdot BN$.

Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).

Each cell of an $n \times n$ board is colored either black or white. A coloring is called [i]good[/i] if every $2 \times 2$ square contains an even number of black cells, and every cross contains an odd number of black cells. Determine all $n \geqslant 3$ such that, in every good coloring, the four corner cells of the board are the same color. [b]Note:[/b] Each $2 \times 2$ square contains exactly $4$ cells of the board. Each cross contains exactly $5$ cells of the board. [asy] size(5cm); // Function to draw a filled square centered at a given position void drawFilledSquare(pair center, real sideLength) { real halfSide = sideLength / 2; fill(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide)), lightgray); draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide))); } // Side length of each square real sideLength = 1; // Coordinates for the cross (left shape) pair[] crossPositions = { (0, 0), (-1, 0), (1, 0), (0, -1), (0, 1) }; // Coordinates for the square (right shape) pair[] squarePositions = { (3, -0.5), (3, 0.5), (4, -0.5), (4, 0.5) }; // Draw the cross for (pair pos : crossPositions) { drawFilledSquare(pos, sideLength); } // Draw the square for (pair pos : squarePositions) { drawFilledSquare(pos, sideLength); } [/asy]

Square $ABCD$ is shown in the diagram below. Points $E$, $F$, and $G$ are on sides $\overline{AB}$, $\overline{BC}$ and $\overline{DA}$, respectively, so that lengths $\overline{BE}$, $\overline{BF}$, and $\overline{DG}$ are equal. Points $H$ and $I$ are the midpoints of segments $\overline{EF}$ and $\overline{CG}$, respectively. Segment $\overline{GJ}$ is the perpendicular bisector of segment $\overline{HI}$. The ratio of the areas of pentagon $AEHJG$ and quadrilateral $CIHF$ can be written as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] draw((0,0)--(50,0)--(50,50)--(0,50)--cycle); label("$A$",(0,50),NW); label("$B$",(50,50),NE); label("$C$",(50,0),SE); label("$D$",(0,0),SW); label("$E$",(0,100/3-1),W); label("$F$",(100/3-1,0),S); label("$G$",(20,50),N); label("$H$",((100/3-1)/2,(100/3-1)/2),SW); label("$I$",(35,25),NE); label("$J$",(((100/3-1)/2+35)/2,((100/3-1)/2+25)/2),S); draw((0,100/3-1)--(100/3-1,0)); draw((20,50)--(50,0)); draw((100/6-1/2,100/6-1/2)--(35,25)); draw((((100/3-1)/2+35)/2,((100/3-1)/2+25)/2)--(20,50)); [/asy]

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

Let $G$ be a simple graph on $n$ vertices with at least one edge, and let us consider those $S:V(G)\to\mathbb R^{\ge 0}$ weighings of the vertices of the graph for which $\sum_{v\in V(G)} S(v)=1$. Furthermore define \[f(G)=\max_S\min_{(v,w)\in E(G)}S(v)S(w),\] where $S$ runs through all possible weighings. Prove that $f(G)=\frac1{n^2}$ if and only if the vertices of $G$ can be covered with a disjoint union of edges and odd cycles. ($V(G)$ denotes the vertices of graph $G$, $E(G)$ denotes the edges of graph $G$.)

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

Given a triangle $ABC$, $BC = 2 \cdot AC$. Point $M$ is the midpoint of side $ BC$ and point $D$ lies on $AB$, with $AD = 2 \cdot BD$. Prove that the lines $AM$ and $MD$ are perpendicular.

Famous French mathematician Pierre Fermat believed that all numbers of the form $F_n = 2^{2^n} + 1$ are prime for all non-negative integers $n$. Indeed, one can check that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ are all prime. a) Prove that $F_5$ is divisible by $641$. (Hence Fermat was wrong.) b) Prove that if $k \ne n$ then $F_k$ and $F_n$ are relatively prime (i.e. they do not have any common divisor except $1$) (Notice: using b) one can prove that there are infinitely many prime numbers)

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?

Let $ABC$ be a triangle with circumcenter $O$, $A$-excenter $I_A$, $B$-excenter $I_B$, and $C$-excenter $I_C$. The incircle of $\Delta ABC$ is tangent to sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Lines $I_BE$ and $I_CF$ intersect at $P$. If the line through $O$ perpendicular to $OP$ passes through $I_A$, prove that $\angle A = 60^\circ$. [i]An excenter is the point of concurrency among one internal angle bisector and two external angle bisectors of a triangle.[/i]

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$. [i]Proposed by Ethan Tan[/i]

Let $A$ be the number of ways in which the set $\{ 1, 2, . . . , n\}$ can be partitioned into non-empty subsets. Let $B$ be the number of ways in which the set $\{ 1, 2, . . . , n, n + 1 \}$ can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that $A = B$.

Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$

Black and white coins are placed on some of the squares of a $418\times 418$ grid. All black coins that are in the same row as any white coin(s) are removed. After that, all white coins that are in the same column as any black coin(s) are removed. If $b$ is the number of black coins remaining and $w$ is the number of remaining white coins, find the remainder when the maximum possible value of $bw$ gets divided by $2007$.

We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.

For a polynomial $P$ and a positive integer $n$, define $P_n$ as the number of positive integer pairs $(a,b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that $P_n \leq 2021$ for all positive integers $n$.

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.