In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius $AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle $DXE$. (Matthew of Kursk)

Consider all $2^{20}$ paths of length $20$ units on the coordinate plane starting from point $(0, 0)$ going only up or right, each one unit at a time. Each such path has a unique [i]bubble space[/i], which is the region of points on the coordinate plane at most one unit away from some point on the path. The average area enclosed by the bubble space of each path, over all $2^{20}$ paths, can be written as $\tfrac{m + n\pi}{p}$ where $m, n, p$ are positive integers and $\gcd(m, n, p) = 1$. Find $m + n + p$.

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$

If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+ ... +(n+1)i^{n}$, where $i=\sqrt{-1}$, equals: $ \textbf{(A)}\ 1+i\qquad\textbf{(B)}\ \frac{1}{2}(n+2) \qquad\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad$ $ \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad\textbf{(E)}\ \frac{1}{8}(n^2+8-4ni) $

In triangle $ABC$, $AB=12$, $BC=17$, and $AC=25$. Distinct points $M$ and $N$ lie on the circumcircle of $ABC$ such that $BM=CM$ and $BN=CN$. If $AM + AN = \tfrac{a\sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $\gcd(a, c) = 1$ and $b$ is not divisible by the square of a prime, compute $100a+10b+c$. [i]Proposed by Michael Tang[/i]

Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$, $$f(xf(y)+y)=yf(x)+f(y).$$

The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours). Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected. [i]V. Dolnikov[/i] [b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].

In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.

Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$) [i]Proposed by Andrew Wu[/i]

Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins. Determine who has a winning strategy depending on $n, k$. [i](Proposed by Fedir Yudin)[/i]

Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.

In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.

Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor .$ [i]Petre Rău[/i]

Three vertices of a parallelogram are $(2,-4),(-2,8),$ and $(12,7.)$ Determine the sum of the three possible x-coordinates of the fourth vertex.

The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon. [asy] size(4cm); dotfactor=3.5; pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,12); B=(0,24); C=(16,36); D=(32,36); E=(48,24); F=(48,12); G=(32,0); H=(16,0); W=origin; X=(0,36); Y=(48,36); Z=(48,0); filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76)); draw(W--X--Y--Z--cycle,linewidth(1.2)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); [/asy]

Prove the identity $\sin^3 a \cos 3a + \cos^3 a \sin 3a=\frac{3}{4}\sin 4a.$

Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.

Zscoder has an simple undirected graph $G$ with $n\ge 3$ vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule: $\bullet$ If the token is currently at vertex $v$ with label $t$, then he can move the token along the edges in $G$ (possibly repeating some edges) exactly $t$ times. After these $t$ moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn. Zscoder claims that he can mark all vertices in $G$ red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must $G$ have, in terms of $n$? [i]Proposed by Yeoh Zi Song[/i]

A circle is inscribed in the triangle $ ABC$ and it's center $I$ and the points of tangency $P, Q, R$ with the sides $BC$, $C A$ and $AB$ are marked, respectively. With a single ruler, build a point $K$ at which the circle passing through the vertices B and $C$ touches (internally) the inscribed circle.

It is known that $(x-y)^3 \vdots 6x^2-2y^2$, where $x,y$ are some integers. Prove that then also $(x+y)^3 \vdots 6x^2-2y^2$.

If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y $ $\forall x,y \in \mathbb{R}$, then determine $g(x+f(y))$.

In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$? [asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = (4,0); C = (4,4); D = (0,4); E = (x,0); F = (4,y); G = (y,4); H = (0,x); I = F + z * dir(225); J = G + z * dir(225); draw(A--B--C--D--A); draw(H--E); draw(J--G^^F--I); draw(rightanglemark(G, J, I), linewidth(.5)); draw(rightanglemark(F, I, E), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$E$", E, S); dot("$F$", F, dir(0)); dot("$G$", G, N); dot("$H$", H, W); dot("$I$", I, SW); dot("$J$", J, SW); [/asy] $\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2$

Let $M$ be the midpoint of the side $BC$ of a acute triangle $ABC$. Incircle of the triangle $ABM$ is tangent to the side $AB$ at the point $D$. Incircle of the triangle $ACM$ is tangent to the side $AC$ at the point $E$. Let $F$ be the such point, that the quadrilateral $DMEF$ is a parallelogram. Prove that $F$ lies on the bisector of $\angle BAC$.

Define $a_1=1$ and $a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$ for $n\in\mathbb{N}$. Find the greatest integer not exceeding $a_{2000}$ and prove your claim.