Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that $$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$

Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.

Let $\Omega_1(O_1, r_1)$ and $\Omega_2(O_2, r_2)$ be two circles that intersect in two points $X, Y$ . Let $A, C$ be the points in which the line $O_1O_2$ cuts the circle $\Omega_1$, and let $B$ be the point in which the circle $\Omega_2$ itnersect the interior of the segment $AC$, and let $M$ be the intersection of the lines $AX$ and $BY$ . Prove that $M$ is the midpoint of the segment $AX$ if and only if $O_1O_2 =\frac12 (r_1 + r_2)$.

For a positive integer $n$ let $\sigma (n)$ be the sum of all its positive divisors. Find all positive integers $n$ such that the number $\frac{\sigma (n)}{n + 1}$ is an integer.

Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.

Let $m \ge 2n$ be two positive integers. Find a closed form for the following expression: $$E(m, n) = \sum_{k=0}^{n} (-1)^k {{m- k} \choose n} { n \choose k}$$

Let $\{f_n\}_{n\ge 1}$ be the Fibonacci sequence, defined by $f_1 = f_2 = 1$, and for all positive integers $n$, $f_{n+2} = f_{n+1} + f_n$. Prove that the following inequality takes place for all positive integers $n$: $${n \choose 1}f_1 +{n \choose 2}f_2+... +{n \choose n}f_n < \frac{(2n + 2)^n}{n!}$$.

Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$ $$f(x + y -xy) = f(x) + f(y) - f(xy).$$

Let $ABC$ be a triangle, and let $C$ be its circumcircle. Let $T$ be the circle tangent to $AB, AC$ and $C$ internally in the points $F, E$ and $D$ respectively. Let $P, Q$ be the intersection points between the line $EF$ and the lines $DB$ and $DC$ respectively. Prove that if $DP = DQ$ then the triangle $ABC$ is isosceles.

Let $N_0$ be the set of all non-negative integers and let $f : N_0 \times N_0 \to [0, +\infty)$ be a function such that $f(a, b) = f(b, a)$ and $$f(a, b) = f(a + 1, b) + f(a, b + 1),$$for all $a, b \in N_0$. Denote by $x_n = f(n, 0)$ for all $n \in N_0$. Prove that for all $n \in N_0$ the following inequality takes place $$2^n x_n \ge x_0.$$

We say that two triangles $T_1$ and $T_2$ are contained one in each other, and we write $T_1 \subset T_2$, if and only if all the points of the triangle $T_1$ lie on the sides or in the interior of the triangle $T_2$. Let $\Delta$ be a triangle of area $S$, and let $\Delta_1 \subset \Delta$ be the largest equilateral triangle with this property, and let $\Delta \subset \Delta_2$ be the smallest equilateral triangle with this property (in terms of areas). Let $S_1, S_2$ be the areas of $\Delta_1, \Delta_2$ respectively. Prove that $S_1S_2 = S^2$. Bonus question: : Does this statement hold for quadrilaterals (and squares)?

Given is a graph $G$. An empty subgraph of $G$ is a subgraph of $G$ with no edges between its vertices. An edge of $G$ is called important if and only if the removal of this edge will increase the size of the maximal empty subgraph. Suppose that two important edges in $G$ have a common endpoint. Prove there exists a cycle of odd length in $G$.

Let $n$ be a positive integer and let $a_n$ be the number of ways to write $n$ as a sum of positive integers, such that any two summands differ by at least $2$. Also, let $b_n$ be the number of ways to write $n$ as a sum of positive integers of the form $5k\pm 1$, $k \in Z$. Prove that $\frac{a_n}{b_n}$ is a constant for all positive integers $n$.

Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.

The sequence $\{x_n\}_n$ is defined as follows: $x_1 = 0$, and for all $n \ge 1$ $$(n + 1)^3 x_{n+1} = 2n^2 (2n + 1)x_n + 2(3n + 1).$$Prove that $\{x_n\}_n$ contains infinitely many integer numbers.

Find all positive integers $a, b, c, d$, such that the following equality takes place for an infinity of positive integers $n$ $$(1^a + 2^a +...+ n^a)^b = (1^c + 2^c +...+ n^c)^d$$

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

If $n>2$ is an integer and $x_1, \ldots ,x_n$ are positive reals such that \[ \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n \]then the following inequality takes place \[ \frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}. \]

Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that $$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$

Let $\Omega$ be the incircle of a triangle $ABC$. Suppose that there exists a circle passing through $B$ and $C$ and tangent to $\Omega$ in $A'$. Suppose the similar points $B'$, $C'$ exist. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.

Let $\{f_n\}_{n \ge 0}$ be the Fibonacci sequence, given by $f_0 = f_1 = 1$, and for all positive integers $n$ the recurrence $f_{n+1} = f_n + f_{n-1}$. Let $a_n = f_{n+1}f_n$ for any non-negative integer $n$, and let $$P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0.$$Prove that for all positive integers $n \ge 3$ the polynomial $P_n(X)$ is irreducible in $Z[X]$.