A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$. Prove that $|B_{n+1}|=3|A_n|$. Vasile Pop

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. Ciprian Manolescu

Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments. Vasile Pop

We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$. Dan Branzei

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. Bogdan Enescu

Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that: i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$ ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$ Radu Todor

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials. Marius Cavachi

Let $ABC$ be an equilateral triangle and $n\ge 2$ be an integer. Denote by $\mathcal{A}$ the set of $n-1$ straight lines which are parallel to $BC$ and divide the surface $[ABC]$ into $n$ polygons having the same area and denote by $\mathcal{P}$ the set of $n-1$ straight lines parallel to $BC$ which divide the surface $[ABC]$ into $n$ polygons having the same perimeter. Prove that the intersection $\mathcal{A} \cap \mathcal{P}$ is empty. Laurentiu Panaitopol

Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that \[ a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n},\] where $ x + A$ denotes the set $ \{x + a \vert a \in A \}$.

Let $n$ be a positive integer and $\mathcal{P}_n$ be the set of integer polynomials of the form $a_0+a_1x+\ldots +a_nx^n$ where $|a_i|\le 2$ for $i=0,1,\ldots ,n$. Find, for each positive integer $k$, the number of elements of the set $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$. Marian Andronache

Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+y)=f(x)u(x)+f(y) \] for all $x,y\in\mathbb{R}$. Vasile Pop

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k + 1\}$, then $ F(0) = F(1) = \ldots = F(k + 1)$.

The lateral surface of a cylinder of revolution is divided by $n-1$ planes parallel to the base and $m$ parallel generators into $mn$ cases $( n\ge 1,m\ge 3)$. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers $k,l$ such that $n+1$ does not divide $k$ and \[ \cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}\] Ciprian Manolescu