RO-TST 1992

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.