Suppose that $ a,b$ are positive integers for which $ A=\frac{a+1}{b}+\frac{b}{a}$ is an integer.Prove that $ A=3$.

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

Prove the following identity for every $ n\in N$: $ \sum_{j+h=n,j\geq h}\frac{(-1)^h2^{j-h}\binom{j}{h}}{j}=\frac{2}{n}$

A sequence $(a_n)$ of positive integers satisfies$(a_m,a_n) = a_{(m,n)}$ for all $m,n$. Prove that there is a unique sequence $(b_n)$ of positive integers such that $a_n = \prod_{d|n} b_d$

Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4. Laurentiu Panaitopol

The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer. Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.

Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders

In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.

Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$. Prove that there is a regular $n$-gon whose all vertices lie on the considered arcs

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$. Andrei Radulescu-Banu