RO-TST 1999

a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11;

b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.

Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$ .

Prove that for any positive integer $n$ , the number $$S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n$$ is the sum of two consecutive perfect squares.

Dorin Andrica

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds $$\frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1.$$

Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that $$x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n).$$ Laurentiu Panaitopol

Prove that for any integer $n$ , $n\geq 3$ , there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that $$b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n .$$ Give an example of two such progressions having at least five terms.

Mihai Baluna

Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and $$x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1.$$ Prove that there exists a positive integer $n$ such that $x_n > 1999!$ .

Ciprian Manolescu

Let $O,A,B,C$ be variable points in the plane such that $OA=4$ , $OB=2\sqrt3$ and $OC=\sqrt {22}$ . Find the maximum value of the area $ABC$ .

Mihai Baluna

Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$ . Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.

Laurentiu Panaitopol

Let $n\geq 3$ and $A_1,A_2,\ldots,A_n$ be points on a circle. Find the largest number of acute triangles that can be considered with vertices in these points.

G. Eckstein

The participants to an international conference are native and foreign scientist. Each native scientist sends a message to a foreign scientist and each foreign scientist sends a message to a native scientist. There are native scientists who did not receive a message.

Prove that there exists a set $S$ of native scientists such that the outer $S$ scientists are exactly those who received messages from those foreign scientists who did not receive messages from scientists belonging to $S$ .

Radu Niculescu

Let $X$ be a set with $n$ elements, and let $A_{1}$ , $A_{2}$ , ..., $A_{m}$ be subsets of $X$ such that:

1. $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$ ;
2. $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$ .

Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$ , the set $A$ does not contain $A_{i}$ .

Alternative formulation. Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$ , such that $|A_{i}\cap A_{j}|\leq 1$ , for every $i\neq j$ . Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$ , such that none of $A_{i}$ 's is a subset of $A$ .

A polyhedron $P$ is given in space. Find whether there exist three edges in $P$ which can be the sides of a triangle. Justify your answer!

Barbu Berceanu