$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$

Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.

In Tigre there are $2024$ islands, some of them connected by a two-way bridge. It is known that it is possible to go from any island to any other island using only the bridges (possibly several of them). In $k$ of the islands there is a flag ($0 \le k \le 2024$). Ana wants to destroy some of the bridges in such a way that after doing so, the following two conditions are met: \\ $\bullet$ If an island has a flag, it is connected to an odd number of islands. \\ $\bullet$ If an island does not have a flag, it is connected to an even number of islands. \\ Determine all values of $k$ for which Ana can always achieve her objective, no matter what the initial bridge configuration is and which islands have a flag.

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?

Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z $). Of all the elegant triangles, which one has the smallest perimeter?

Find the number of ordered quadruples of positive integers $(a,b,c, d)$ such that $ab + cd = 10$.

A robot starts at the origin of the Cartesian plane. At each of $10$ steps, he decides to move $ 1$ unit in any of the following directions: left, right, up, or down, each with equal probability. After $10$ steps, the probability that the robot is at the origin is $\frac{n}{4^{10}}$ . Find$ n$

In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

$a)$ Prove that for all positive integers $n \geq 3$ holds: $$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent $b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers

Let $ABCD$ be a fixed convex quadrilateral. Say a point $K$ is [i]pastanaga[/i] if there's a rectangle $PQRS$ centered at $K$ such that $A\in PQ, B\in QR, C\in RS, D\in SP$. Prove there exists a circle $\omega$ depending only on $ABCD$ that contains all pastanaga points.

In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively. Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively. Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$. Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.

The length of rectangle R is $ 10$ percent more than the side of square S. The width of the rectangle is $ 10$ percent less than the side of the square. The ratio of the areas, R:S, is: $ \textbf{(A)}\ 99: 100 \qquad \textbf{(B)}\ 101: 100 \qquad \textbf{(C)}\ 1: 1 \qquad \textbf{(D)}\ 199: 200 \qquad \textbf{(E)}\ 201: 200$

Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.

Determine the maximum value attained by \[\dfrac{x^4-x^2}{x^6+2x^3-1}\] over real numbers $x>1.$

We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$): \[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\] [i]Proposed by D. Malec, C. Tompkins[/i]

A sequence $a_n$ is defined such that $a_0 =\frac{1 + \sqrt3}{2}$ and $a_{n+1} =\sqrt{a_n}$ for $n \ge 0$. Evaluate $$\prod_{k=0}^{\infty} 1 - a_k + a^2_k$$

Find the complex numbers $ z$ for which the series \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\] converges and find its sum.

In $\triangle ABC$ let $D$ and $E$ be points on $AB$ and $AC$ respectively. The circumcircle of $\triangle CDE$ meets $AB$ again at $F$, and the circumcircle of $\triangle ACD$ meets $BC$ again at $G$. Show that if the circumcircles of $DFG$ and $ADE$ meet at $H$, then the three lines $AG$, $BE$, and $DH$ concur. Proposed by [i]Oron Wang[/i] inspired by [i]Tiger Zhang[/i]

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

[b]a.[/b] There are $5$ identical paper triangles on the table. Each can be moved in any direction parallel to itself (i.e., without rotating it). Is it true that then any one of them can be covered by the $4$ others? [b]b.[/b] There are $5$ identical equilateral paper triangles on the table. Each can be moved in any direction parallel to itself. Prove that any one of them can be covered by the $4$ others in this way.

Determine the least real number $k$ such that the inequality $$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$ holds for all real numbers $a,b,c$. [i]Proposed by Mohammad Jafari[/i]

Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.

On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.