Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value $t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$

Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

There are $22$ chairs in a round table. Find the minimum n such that for any group of $n$ people sitting in the table, we always can find two people with exactly $2$ or $8$ chairs between them. (Le Anh Vinh)

Consider the paths of length $16$ that go from the lower left corner to the upper right corner of an $8\times 8$ grid. Find the number of such paths that change direction exactly $4$ times.

In a group of $n > 20$ people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of $n$ and $k,$ this group has a peculiar property: If any $20$ people are removed from the group, the number of pairs of people that know each other is at most $\frac{n-k}{n}$ times that of the original group of people. (a) If $k = 41,$ for what positive integers $n$ could such a group exist? (b) If $k = 39,$ for what positive integers $n$ could such a group exist?

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5? $\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)? (A) From A to B (B) From B to C only (C) From B to D (D) From C to D only (E) From D to E

Let $n,m$ be positive integers such that $n<m$ and $a_1, a_2, ..., a_m$ be different real numbers. (a) Find all polynomials $P$ with real coefficients and degree at most $n$ such that: $|P(a_i)-P(a_j)|=|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$. (b) If $n,m\ge 2$ does there exist a polynomial $Q$ with real coefficients and degree $n$ such that: $|Q(a_i)-Q(a_j)|<|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$ Edit: See #3

The points of this lattice $4\times 4 = 16$ points can be vertices of squares. [asy] unitsize(1 cm); int i, j; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((1,1)--(2,2)--(1,3)--(0,2)--cycle); for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy] Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

Let $ABC$ be a triangle. Circle $\Omega$ passes through points $B$ and $C$. Circle $\omega$ is tangent internally to $\Omega$ and also to sides $AB$ and $AC$ at $T,~ P,$ and $Q$, respectively. Let $M$ be midpoint of arc $\widehat{BC}$ (containing T) of $\Omega$. Prove that lines $P Q,~ BC,$ and $MT$ are concurrent.

Let $n>5$ be an integer. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1<i<n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the convex hull of the points at that moment has $v(i)$ vertices. Finally, he writes down $v(n-2) = 3$. Find the greatest possible value that the expression $$|v(1)-v(2)|+ |v(2)-v(3)| + \ldots + |v(n-3)-v(n-2)|$$ can obtain among all possible initial configurations of $n$ points and all possible Tom's moves. [i]Remark[/i]. A convex hull of a finite set of points in the plane is the smallest convex polygon containing all the points of the set (inside it or on the boundary). [i]Ivan Novak, Namik Agić[/i]

In a parallelogram $\Box ABCD$ $(AB < BC)$ The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$. The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$. Let $S$ = $PQ$ $\cap$ $AD$ $U$ = $AR$ $\cap$ $CS$ $T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$ Prove that $AT, BU, PQ$ are concurrent

In a triangle $ABC$, the points $A_1, B_1, C_1$ are symmetric to $A, B,C$ with respect to $B,C, A$, respectively. Given the points $A_1, B_1,C_1$ reconstruct the triangle $ABC$.

Finite set $A$ has such property: every six its distinct elements’ sum isn’t divisible by $6$. Does there exist such set $A$ consisting of $11$ distinct natural numbers?

$10000$ nonzero digits are written in a $100$-by-$100$ table, one digit per cell. From left to right, each row forms a $100$-digit integer. From top to bottom, each column forms a $100$-digit integer. So the rows and columns form $200$ integers (each with $100$ digits), not necessarily distinct. Prove that if at least $199$ of these $200$ numbers are divisible by $2013$, then all of them are divisible by $2013$.

An $8\times 8$ square board is divided into $64$ unit squares. A ’skew-diagonal’ of the board is a set of $8$ unit squares no two of which are in the same row or same column. Checkers are placed in some of the unit squares so that ’each skew-diagonal contains exactly two squares occupied by checkers’. Prove that there exist two rows or two columns which contain all the checkers.