Let $ a_n = 1 + 2 + ... + n$ for every $ n \ge 1$; the numbers $ a_n$ are called triangular. Prove that if $ 2a_m = a_n$ then $ a_{2m - n}$ is a perfect square.

Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.

Let $ b$ be an even positive integer for which there exists a natural number n such that $ n>1$ and $ \frac{b^n-1}{b-1}$ is a perfect square. Prove that $ b$ is divisible by 8.

The Fibonacci sequence is determined by conditions $ F_0 = 0, F1 = 1$, and $ F_k=F_{k-1}+F_{k-2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) = a_mx^m +. . .+ a_1x+ a_0$ be a polynomial that satisfies the following two conditions: (1) $ P(F_n) = F_{n}^{2}$ ; (2) $ P(F_k) = P(F_{k-1}) + P(F_{k-2}$ for all $ k \ge 2$. Find the sum of the coefficients of P.

Let $n$ be a fixed natural number. The maze is a grid of dimensions $n \times n$, with a gate to the sky on one of the squares and some adjacent squares with partitions separated from each other so that it is still possible to move from one square to another. The program is in the UP, DOWN, RIGHT, LEFT final sequence, With each command, the Creature moves from its current square to the corresponding neighboring square, unless the partition or the outer boundary of the labyrinth prevents execution of the command (otherwise it does nothing), upon entering the gate, the Creature moves on to heaven. God creates a program, then Satan creates a labyrinth and places it on a square. Prove that God can make such a program that, independently of Satan's labyrinth and selected from the source square, the Creature always reaches heaven by following this program.

A Bluetooth device can connect to any other Bluetooth device that is not more than $10$ meters from him. A piconet is called a bluetooth network consisting of one master and a plurality of connected slaves. What is the greatest number of slaves, what can be on the pickup provided that all devices are on the same level and all slaves are out of range of each other?

Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of some $ n$ consecutive integers is a perfect square?

Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.

Find all positive integers n such that one can write an integer 1 to $ n^2$ into each unit square of a $ n^2 \times n^2$ table in such a way that, in each row, each column and each $ n \times n$ block of unit squares, each number 1 to $ n^2$ occurs exactly once.

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?