For an acute triangle $ ABC$ prove the inequality: $ \sum_{cyclic} \frac{m_a^2}{-a^2+b^2+c^2}\ge \frac{9}{4}$ where $ m_a,m_b,m_c$ are lengths of corresponding medians.

Let $ x,y,z$ be positive real numbers such that $ x+2y+3z=\frac{11}{12}$. Prove the inequality $ 6(3xy+4xz+2yz)+6x+3y+4z+72xyz\le \frac{107}{18}$.

Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called summable if $ \sum_{i=1}^n \frac{1}{x_i}=1$. Prove that for every $ n\ge 3$ there always exists a summable set which consists of $ n$ elements such that the biggest element is: a) bigger than $ 2^{2n-2}$ b) smaller than $ n^2$

Determine the biggest possible value of $ m$ for which the equation $ 2005x + 2007y = m$ has unique solution in natural numbers.

Determine all pairs $ (m,n)$ of natural numbers for which $ m^2=nk+2$ where $ k=\overline{n1}$. EDIT. It has been discovered the correct statement is with $ k=\overline{1n}$.

Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n-1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n-1} \rfloor$.

Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}=11(a+b+c+d)^2$.

Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

Let $ ABCD$ be a trapezoid with $ AB\parallel CD,AB>CD$ and $ \angle{A} + \angle{B} = 90^\circ$. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.

Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB=2CD$. Let $ \{Q\}=AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.

Circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ intersect at $ A$ and $ B$. Let $ M\in AB$. A line through $ M$ (different from $ AB$) cuts circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ at $ Z,D,E,C$ respectively such that $ D,E\in ZC$. Perpendiculars at $ B$ to the lines $ EB,ZB$ and $ AD$ respectively cut circle $ \mathcal{C}_2$ in $ F,K$ and $ N$. Prove that $ KF=NC$.

Let $ ABC$ be an equilateral triangle of center $ O$, and $ M\in BC$. Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$.

Let $ A$ be a subset of the set $ \{1, 2,\ldots,2006\}$, consisting of $ 1004$ elements. Prove that there exist $ 3$ distinct numbers $ a,b,c\in A$ such that $ gcd(a,b)$: a) divides $ c$ b) doesn't divide $ c$

Let $ n\ge 5$ be a positive integer. Prove that the set $ \{1,2,\ldots,n\}$ can be partitioned into two non-zero subsets $ S_n$ and $ P_n$ such that the sum of elements in $ S_n$ is equal to the product of elements in $ P_n$.

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.