The square $DEAF$ is constructed inside the $30^o-60^o-90^o$ triangle $ABC$, with the hypotenuse $BC = 4$, $D$ on side $BC$, E on side $AC$, and F on side $AB$. What is the side length of the square?

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$

The Macedonian Mathematical Olympiad is held in two rooms numbered $1$ and $2$. At the beginning all of the competitors enter room No. $1$. The final arrangement of the competitors to the rooms is obtained in the following way: a list with the names of a few of the competitors is read aloud; after a name is read, the corresponding competitor and all of his/her acquaintances from the rest of the competitors change the room in which they currently are. Hence, to each list of names corresponds one final arrangement of the competitors to the rooms. Show that the total number of possible final arrangements is not equal to $2009$ (acquaintance between competitors is a symmetrical relation).

We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.

Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$

Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$. Call the resulting matrix $A$. Determine the probability that $A^2 = O$, as a function of $n$.

Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.

The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?

What is the value of $\frac{(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})}{\sqrt{(1-\frac{1}{3^2})(1-\frac{1}{5^2})(1-\frac{1}{7^2})}}?$ $\textbf{(A) }\sqrt{3} \qquad \textbf{(B) }2 \qquad \textbf{(C) }\sqrt{15} \qquad \textbf{(D) }4 \qquad \textbf{(E) }\sqrt{105}$

Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

Let $n$ be a positive integer with $n \geq 2$, and $0<a_{1}, a_{2},...,a_{n}< 1$. Find the maximum value of the sum $\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$ where $a_{n+1}=a_{1}$

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.

Let $\omega$ be the circumcircle of the triangle $ABC$ with $\angle B = 3\angle C$. The internal angle bisector of $\angle A$, intersects $\omega$ and $BC$ at $M$ and $D$, respectively. Point $E$ lies on the extension of the line $MC$ from $M$ such that $ME$ is equal to the radius of $\omega$. Prove that circumcircles of triangles $ACE$ and $BDM$ are tangent. [i]Proposed by Mehran Talaei - Iran[/i]

A kite with $AB = BC$ and $AD = CD$ has diagonals which satisfy $AC = 80$ and $BD = 71$. Let $AC$ and $BD$ intersect at a point $O$. Find the area of the quadrilateral formed by the circumcenters of $ABO$, $BCO$, $CDO$, and $ADO$.

Let $0<a_1<a_2< \dots < a_{mn+1}$ be $mn+1$ integers. Prove that you can select either $m+1$ of them no one of which divides any other, or $n+1$ of them each dividing the following one.

Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that \[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\] Find $\tan^{2}\left(\measuredangle AEP\right)$.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$. [i]Proposed by Japan[/i]

Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.

Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?