Let $ABC$ be an acute, non isosceles triangle, $AX, BY, CZ$ are the altitudes with $X, Y, Z$ belong to $BC, CA,AB$ respectively. Respectively denote $(O_1), (O_2), (O_3)$ as the circumcircles of triangles $AY Z, BZX, CX Y$ . Suppose that $(K)$ is a circle that internal tangent to $(O_1), (O_2), (O_3)$. Prove that $(K)$ is tangent to circumcircle of triangle $ABC$.

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Prove that in a triangle $Sum of medians >\frac{3}{4}(perimeter of triangle )$

A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.

Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$. [i]Milan Haiman[/i]

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 22 \times 23$ in order to make it a perfect square?

Let $M$ be a $2014\times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F\in\mathcal{F}(M)$ such that $\det(M + F) \ne 0$.

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

The positive integers are colored with black and white such that: - There exists a bijection from the black numbers to the white numbers, - The sum of three black numbers is a black number, and - The sum of three white numbers is a white number. Find the number of possible colorings that satisfies the above conditions.

Prove that for all positive real numbers $x, y$ holds the inequality $$x^4 + y^3 + x^2 + y + 1 > \frac92 xy.$$

Consider $v,w$ two distinct non-zero complex numbers. Prove that \[|zw+\bar{w}|\le |zv+\bar{v}|,\] for any $z\in\mathbb{C},|z|=1$, if and only if there exists $k\in [-1,1]$ such that $w=kv$. [i]Dan Marinescu[/i]

A large flat plate of glass is suspended $\sqrt{2/3}$ units above a large flat plate of wood. (The glass is infinitely thin and causes no funny refractive effects.) A point source of light is suspended $\sqrt{6}$ units above the glass plate. An object rests on the glass plate of the following description. Its base is an isosceles trapezoid $ABCD$ with $AB \parallel DC$, $AB = AD = BC = 1$, and $DC = 2$. The point source of light is directly above the midpoint of $CD$. The object’s upper face is a triangle $EF G$ with $EF = 2$, $EG = F G =\sqrt3$. $G$ and $AB$ lie on opposite sides of the rectangle $EF CD$. The other sides of the object are $EA = ED = 1$, $F B = F C = 1$, and $GD = GC = 2$. Compute the area of the shadow that the object casts on the wood plate.

Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty. Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$

Let triangle $ABC$ be a triangle right at $B$. The inscribed circle is tangent to sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $CF$ intersect the circle at the point $P$. If $\angle APB=90^{\circ}$, find the value of $\dfrac{CP+CD}{PF}$. [i](tatari/nightmare)[/i]

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

$2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)