RO-TST 1987

Let $a,b,c$ be distinct real numbers such that $a+b+c>0$ . Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$ . Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.

Mircea Becheanu

Find all positive integers $A$ which can be represented in the form: $$A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right)$$ where $m\geq n\geq p \geq 1$ are integer numbers.

Ioan Bogdan

Let $A$ be the set $A = \{ 1,2, \ldots, n\}$ . Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$ , $x+y$ cannot be divisible by $x-y$ .

Mircea Lascu, Dorel Mihet

Let $P(X) = a_{n}X^{n} + a_{n - 1}X^{n - 1} + \ldots + a_{1}X + a_{0}$ be a real polynomial of degree $n$ . Suppose $n$ is an even number and:

a) $a_{0} > 0$ , $a_{n} > 0$ ;

b) $a_{1}^{2} + a_{2}^{2} + \ldots + a_{n - 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n - 1}$ .

Prove that $P(x)\geq 0$ for all real values $x$ .

Laurentiu Panaitopol

Let $A$ be the set $\{1,2,\ldots,n\}$ , $n\geq 2$ . Find the least number $n$ for which there exist permutations $\alpha$ , $\beta$ , $\gamma$ , $\delta$ of the set $A$ with the property: $$\sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) .$$ Marcel Chirita

The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons.

Gabriel Nagy

Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$ .

Let $ABCD$ be a square and $a$ be the length of his edges. The segments $AE$ and $CF$ are perpendicular on the square's plane in the same half-space and they have the length $AE=a$ , $CF=b$ where $a<b<a\sqrt 3$ . If $K$ denoted the set of the interior points of the square $ABCD$ determine $\min_{M\in K} \left( \max ( EM, FM ) \right)$ and $\max_{M\in K} \left( \min (EM,FM) \right)$ .

Octavian Stanasila

Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have $$\sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0.$$ Octavian Stanasila

Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$ . Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$ .

Laurentiu Panaitopol

Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$ . For a given positive integer $n$ , $n\geq 2$ consider the system of equations: $$P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 .$$ We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$ , $x_1=\lambda y_1$ , $\ldots$ , $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?

Mircea Becheanu