Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$Find all possible values of $a$.

In an acute-angled triangle $ABC, E$ and $F$ are the feet of the altitudes from $B$ and $C$, and $G$ and $H$ are the projections of $B$ and $C$ on $EF$, respectively. Prove that $HE = FG$.

Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.

Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality \[ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} \]

Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.

Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Suppose that there are circles $\omega_2,...,\omega_7$ such that: (a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$, (b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$. Prove that $\omega_1 = \omega_7$.

Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.