Suppose that$ f : N \to N$ is an increasing function such that $f(f(n)) = 3n$ for all $n$. Find $f(1992)$.

For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere. Dan Brânzei

Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$. If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.

Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.

Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct. a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $; b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds. Șerban Buzețeanu

Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.

Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.

Let $m,n$ be positive integers and $p$ be a prime number. Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.

Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be the sequence of positive integers defined by $a_{n+1}=na_{n}+1$ and $b_{n+1}=nb_{n}-1$ for $n\geq 1$. Show that the two sequence cannot have infinitely many common terms. Laurentiu Panaitopol

Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}$

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$. Laurențiu Panaitopol and Ioan Tomescu

In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$

In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.