Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.

On a board with $3$ columns and $4$ rows, each of the $12$ squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?

Let $f: N^+ \to N^+$ be a function that satisfies that $$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$ Prove that $$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$

Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.

Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.

Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$

Let $a \star b = ab + a + b$ for all integers $a$ and $b$. Evaluate $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$.

Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of $$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$

The polynomial $P(x) = 3x^3 -2x^2 +ax+b$ has roots $\sin^2 \theta$, $\cos^2 \theta$, and $\sin \theta \cos\theta$ for some angle $\theta$. Compute $P(1)$.

For a set \(S\) of positive integers and a positive integer \(n\), consider the game of [i]\((n,S)\)-nim[/i], which is as follows. A pile starts with \(n\) watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of \(S\). The last player to move wins. Let \(f(S)\) denote the set of positive integers \(n\) for which Deric has a winning strategy in \((n,S)\)-nim. Let \(T\) be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\] be eventually constant? [i]Proposed by Brandon Wang and Edward Wan[/i]

Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there?

Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$

An infinite sequence is given by $x_1=2, x_2=7, x_{n+1} = 4x_n - x_{n-1}$ for all $n \geq 2$. Does there exist a perfect square in this sequence? [hide="Remark"]During the test the initial value of $x_1$ was given as $1$, thus the problem was not graded[/hide]

Find the largest and smallest value of the function $$y=\sqrt{7+5\cos x}-\cos x.$$

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

Let $A$ be a fixed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a fixed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some fixed circle not depending on $BC$.

The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$? [asy]defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$

Let $L$ be a figure made of $3$ squares, a right isosceles triangle and a quarter circle (all unit sized) as shown below: [img]https://wiki-images.artofproblemsolving.com//f/f9/Weirwiueripo.png[/img] Prove that any $18$ points in the plane can be covered with copies of $L$, which don't overlap (copies of $L$ may be rotated or flipped)

Positive real numbers $a_1, \dots, a_k, b_1, \dots, b_k$ are given. Let $A = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i$. Prove the inequality \[ \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}. \]

Consider the circle $\gamma$ with center at point $(0,3)$ and radius $3$, and a line $r$ parallel to the axis $Ox$ at a distance $3$ from the origin. A variable line through the origin meets $\gamma$ at point $M$ and $r$ at point $P$. Find the locus of the intersection point of the lines through $M$ and $P$ parallel to $Ox$ and $Oy$ respectively.

$a_1 + a_2 + ... + a_n = 0$, for some $k$ we have $a_j \le 0$ for $j \le k$ and $a_j \ge 0$ for $j > k$. If ai are not all $0$, show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$.

Given $n \in N$, let $A$ be a family of subsets of $\{1,2,...,n\}$. If for every two sets $B,C \in A$ the set $(B \cup C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .

Find all triples $(x,y, z)$ of real numbers such that $x^2 + y^2 + z^2 + 1 = xy + yz + zx + |x - 2y + z|$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]