Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.

Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$. Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$, and let $K$ and $ L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.

Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy? [i]Note: an empty graph is a graph with no edges.[/i]

Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order.

We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$. (a) Prove that for every natural $n$ holds: $x_n > \frac12$ (b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.

How many positive integers less than 10,000 have at most two different digits?

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

Let $m$ be a positive integer. Find the number of real solutions of the equation $$|\sum_{k=0}^{m} \binom{2m}{2k}x^k|=|x-1|^m$$

9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$. ([i]A. Badzyan[/i])

Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times? [i]Proposed by Matthew Weiss

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

The set of $ x$-values satisfying the inequality $ 2 \leq |x\minus{}1| \leq 5$ is: $ \textbf{(A)}\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad \textbf{(B)}\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad \textbf{(C)}\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad \textbf{(D)}\ \minus{}1 \leq x \leq 3 \qquad \textbf{(E)}\ \minus{}4 \leq x \leq 6$

Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.

Let $ABC$ be a scalene triangle. The tangents at the perpendicular foot dropped from $A$ on the line $BC$ and the midpoint of the side $BC$ to the nine-point circle meet at the point $A'$\,; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent. [i]Gazeta Matematica[/i]

For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

Consider the functions $f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}$ and the binary operation $*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ defined as \[x*y=f(x)+g(y)+h(x)\cdot|x-y|,\]for all $x,y\in\mathbb{R}_{\geqslant 0}$. Suppose that $(\mathbb{R}_{\geqslant 0},*)$ is a commutative monoid. Determine the functions $f,g,h$.

Three digits are selected at random and without replacement from the set of digits $0$ through $9$. What is the probability that the three digits can be arranged to form a multiple of $5$? $\text{(A) }\frac{17}{90}\qquad\text{(B) }\frac{7}{15}\qquad\text{(C) }\frac{1}{2}\qquad\text{(D) }\frac{8}{15}\qquad\text{(E) }\frac{1}{5}$

In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AX B = 50^o$, $\angle CAD = 51^o$, $\angle AED = 80^o$. Find the value of angle $\angle AXC$.

Find the minimum value of $ 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15$ for real numbers $ x$, $ y$, $ z$.

Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.

Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$? (B. Frenkin)