Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle. (Anton Trygub)

Evaluate: \[\sum^{10}_{x=2} \dfrac{2}{x(x^2-1)}\] .

Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle [img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]

Solve in real numbers the system: $$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

[color=darkblue]Let $ X$ be a group of people, where any two people are friends or enemies. Each pair of friends from $ X$ doesn't have any common friends, and any two enemies have exactly two common friends. Prove that each person from $ X$ has the same number of friends as others.[/color]

Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$. [i]Proposed by Jeff Lin[/i]

In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. Prove that each participant has lost a different number of games.

If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x^2 +f(y))=y+xf(x)$ for all $x,y \in \mathbb{R}$, find $f(x)$.

For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence \[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \] to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).

An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.

Let $ABC$ be a triangle with $AB = 130$, $BC = 140$, $CA = 150$. Let $G$, $H$, $I$, $O$, $N$, $K$, $L$ be the centroid, orthocenter, incenter, circumenter, nine-point center, the symmedian point, and the de Longchamps point. Let $D$, $E$, $F$ be the feet of the altitudes of $A$, $B$, $C$ on the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$. Let $X$, $Y$, $Z$ be the $A$, $B$, $C$ excenters and let $U$, $V$, $W$ denote the midpoints of $\overline{IX}$, $\overline{IY}$, $\overline{IZ}$ (i.e. the midpoints of the arcs of $(ABC)$.) Let $R$, $S$, $T$ denote the isogonal conjugates of the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $P$ and $Q$ denote the images of $G$ and $H$ under an inversion around the circumcircle of $ABC$ followed by a dilation at $O$ with factor $\frac 12$, and denote by $M$ the midpoint of $\overline{PQ}$. Then let $J$ be a point such that $JKLM$ is a parallelogram. Find the perimeter of the convex hull of the self-intersecting $17$-gon $LETSTRADEBITCOINS$ to the nearest integer. A diagram has been included but may not be to scale. [asy] size(6cm); import olympiad; import cse5; pair A = dir(110); pair B = dir(210); pair C = dir(330); pair D = foot(A,B,C); pair E = foot(B,C,A); pair F = foot(C,A,B); pair G = centroid(A,B,C); pair H = orthocenter(A,B,C); pair I = incenter(A,B,C); pair isocon(pair targ) { return extension(A,2*foot(targ,I,A)-targ, C,2*foot(targ,I,C)-targ); } pair O = circumcenter(A,B,C); pair K = isocon(G); pair N = midpoint(O--H); pair U = extension(O,midpoint(B--C),A,I); pair V = extension(O,midpoint(C--A),B,I); pair W = extension(O,midpoint(A--B),C,I); pair X = -I + 2*U; pair Y = -I + 2*V; pair Z = -I + 2*W; pair R = isocon(midpoint(A--D)); pair S = isocon(midpoint(B--E)); pair T = isocon(midpoint(C--F)); pair L = 2*H-O; pair P = 0.5/conj(G); pair Q = 0.5/conj(H); pair M = midpoint(P--Q); pair J = K+M-L; draw(A--B--C--cycle); void draw_cevians(pair target) { draw(A--extension(A,target,B,C)); draw(B--extension(B,target,C,A)); draw(C--extension(C,target,A,B)); } draw_cevians(H); draw_cevians(G); draw_cevians(I); draw(unitcircle); draw(circumcircle(D,E,F)); draw(O--P); draw(O--Q); draw(P--Q); draw(CP(X,foot(X,B,C))); draw(CP(Y,foot(Y,C,A))); draw(CP(Z,foot(Z,A,B))); draw(J--K--L--M); draw(X--Y--Z--cycle); draw(A--X); draw(B--Y); draw(C--Z); draw(A--foot(X,A,B)); draw(A--foot(X,A,C)); draw(B--foot(Y,B,C)); draw(B--foot(Y,B,A)); draw(C--foot(Z,C,A)); draw(C--foot(Z,C,B)); pen p = black; dot(A, p); dot(B, p); dot(C, p); dot(D, p); dot(E, p); dot(F, p); dot(G, p); dot(H, p); dot(I, p); dot(J, p); dot(K, p); dot(L, p); dot(M, p); dot(N, p); dot(O, p); dot(P, p); dot(Q, p); dot(R, p); dot(S, p); dot(T, p); dot(U, p); dot(V, p); dot(W, p); dot(X, p); dot(Y, p); dot(Z, p); [/asy]

Let $x$ be the length of one side of a triangle and let $y$ be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle.

Find all primes $p,q$ such that $p^3-q^7=p-q$.

Prove the existence of a unique sequence $\{u_n\} \ (n = 0, 1, 2 \ldots )$ of positive integers such that \[u_n^2 = \sum_{r=0}^n \binom{n+r}{r} u_{n-r} \qquad \text{for all } n \geq 0\]

If $a, b, c, d$ are reals with $a \ge b \ge c \ge d \ge 0$ and $b(b-a)+c(c-b)+d(d-c) \le 2 - \frac{a^2}{2}$, find the minimum value of the expression \begin{align*}\frac{1}{b+2006c-2006d}+\frac{1}{a+2006b-2006c-d} + \frac{1}{2007a-2006b-c+d} + \frac{1}{a-b+c+2006d}.\end{align*}

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

Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.

Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called [i]colorful[/i] if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.

From a $8\times 7$ rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has $14$ unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$.

Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, ex-center in angle $A$ is $J$. Denote $D$ as the tangent point of $(I)$ on $BC$ and the angle bisector of angle $A$ cuts $BC$, $(O)$ respectively at $E, F$. The circle $(DEF )$ meets $(O)$ again at $T$. Prove that $AT$ passes through an intersection of $(J)$ and $(DEF )$.

Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that \[|E(\{0,1,3\})| \geq |E(\{0,1,2\})|\] with equality if and only if $p = 5$.

Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by $2014$, divide by $2014$. The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$, $a\ge 2015$ such that all numbers of the form $an+b$ will not be the starting integer, where $n$ is any positive integer. Find the minimum number of starting integer where Navi wins.