RO-TST 2005

Solve the equation $3^x=2^xy+1$ in positive integers.

Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$ . Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$ .

Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$ .

Calin Popescu

Prove that in any convex polygon with $4n+2$ sides ( $n\geq 1$ ) there exist two consecutive sides which form a triangle of area at most $\frac 1{6n}$ of the area of the polygon.

Let $m,n$ be co-prime integers, such that $m$ is even and $n$ is odd. Prove that the following expression does not depend on the values of $m$ and $n$ : $$\frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} .$$

Bogdan Enescu

A sequence of real numbers $\{a_n\}_n$ is called a bs sequence if $a_n = |a_{n+1} - a_{n+2}|$ , for all $n\geq 0$ . Prove that a bs sequence is bounded if and only if the function $f$ given by $f(n,k)=a_na_k(a_n-a_k)$ , for all $n,k\geq 0$ is the null function.

Mihai Baluna - ISL 2004

Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$ , $A_2'$ , $A_4'$ on the circle, such that $$A_0A_0' \parallel A_2A_4, A_2A_2' \parallel A_4A_0, A_4A_4' \parallel A_2A_0 .$$

Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$ , the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$ .

Prove that if the lines $A_0A_3$ , $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$ , $A_4A_1'$ and $A_2A_5'$ are also concurrent.

Let $ABC$ be a triangle, and let $D$ , $E$ , $F$ be 3 points on the sides $BC$ , $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$ , $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$ . Prove that $D$ , $E$ , $F$ are the midpoints of the sides of the triangle $ABC$ .

Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$ .

Let $a\in\mathbb{R}-\{0\}$ . Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$ .

Dan Schwartz

On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.

Dan Schwartz

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that $$\sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 .$$

Călin Popescu

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$ , between $c_n$ and $c_{n+1}$ , the infinite sequence thus obtained does not represent the fractional part of a rational number.

b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$ .

Dan Schwartz

On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}.

Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$ , $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$ , the inequality $$\sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r$$ is true for all $r\geq \rho (n)$ .

Let $\mathbb{N}=\{1,2,\ldots\}$ . Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$ .

We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge.

Prove that for all integers $n\geq 3$ there exists a face of the polyhedra with a number of sides not divisible by $n$ .