Let $I$ and $H$ be the incenter and orthocenter of an acute triangle $ABC$. $M$ is the midpoint of arc $AC$ of circumcircle of triangle $ABC$ which does not contain point $B$. If $MI=MH$, find the measure of angle $\angle ABC$. [I]Proposed by F. Bakharev[/i]

Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$? $ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reflected about the $ x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("$A$",D2(0,big),NW); label("$B$",D2(0,0),SW); label("$C$",D2(big,0),SW); label("$D$",D2(big,big),N); label("$E$",D2(big+small,0),SE); label("$F$",D2(big+small,small),NE); label("$G$",D2(big,small),NE); [/asy]

If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]

Initially, there are $14$ numbers written on the board - zeros and ones. Every minute, Anton chooses half of the numbers on the board and adds $1$ to each of them, while Mykhailo multiplies all the other numbers by $8$. At some point (possibly initially), all the numbers on the board become equal. How many ones could have been on the board initially? [i]Proposed by Oleksii Masalitin[/i]

If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.

For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \] Show that, for $0\le a,b,c,d\le 99$ \[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ . Proposed by Alireza Dadgarnia

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

If the radius of a circle is increased by $ 1$ unit, the ratio of the new circumference to the new diameter is: $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ \frac{2 \pi \plus{} 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2 \pi \minus{} 1}{2} \qquad \textbf{(E)}\ \pi \minus{} 2$

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined. Determine the number of ways to select positive integers $a$, $b$, $c$ such that [list] [*] the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and [*] the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$. [/list]

Numbers $1,2,\ldots,50$ are written on the board. Anya does the following operation: removes the numbers $a$ and $b$ from the board and writes their sum - $a+b$, after which also notes down the number $ab(a+b)$. After $49$ of this operations only one number was left on the board. Anya summed up all the $49$ numbers in her notes and got $S$. a) Prove that $S$ does not depend on the order of Anya's actions. b) Calculate $S$.

Given a square with side $10$. Cut it into $100$ congruent quadrilaterals such that each of them is inscribed into a circle with diameter $\sqrt{3}$. [i](5 points)[/i] [i]Ilya Bogdanov[/i]

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$. [i]Proposed by Evan Chen[/i]

$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]