For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.

Let $A$ be a finite set of positive real numbers satisfying the property: For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity. Show that the product of all elements in $A$ is equal to $1$.

To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares: $P$ and another $2$ unit squares with a common vertex, as shown below: Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).

Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

Let $ABCD$ be a quadrilateral. The diagonals $AC$ and $BD$ are perpendicular at point $O$. The perpendiculars from $O$ on the sides of the quadrilateral meet $AB, BC, CD, DA$ at $M, N, P, Q$, respectively, and meet again $CD, DA, AB, BC$ at $M', N', P', Q'$, respectively. Prove that points $M, N, P, Q, M', N', P', Q'$ are concyclic. Cosmin Pohoata

Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that: $$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$

Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ . Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .

Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.

The plane is divided into a net of equilateral triangles of side length $1$, with disjoint interiors. A checker is placed initialy inside a triangle. The checker can be moved into another triangle sharing a common vertex (with the triangle hosting the checker) and having the opposite sides (with respect to this vertex) parallel. A path consists in a finite sequence of moves. Prove that there is no path between two triangles sharing a common side.

Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such that $XZ + Y Z \le 1$

Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds: $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$

A positive integer is called saturated i f any prime factor occurs at a power greater than or equal to $2$ in its factorisation. For example, numbers $8 = 2^3$ and $9 = 3^2$ are saturated, moreover, they are consecutive. Prove that there exist infinitely many saturated consecutive numbers.

Let $ABC$ be a triangle and $A_1$ the foot of the internal bisector of angle $BAC$. Consider $d_A$ the perpendicular line from $A_1$ on $BC$. Define analogously the lines $d_B$ and $d_C$. Prove that lines $d_A, d_B$ and $d_C$ are concurrent if and only if triangle $ABC$ is isosceles.

Show that there exist (at least) a rearrangement $a_0, a_1, a_2,..., a_{63}$ of the numbers $0,1, 2,..., 63$, such that $a_i - a_j \ne a_j - a_k$, for any $i < j < k \in \{0,1, 2,..., 63\}$.