RO-TST 1988

Consider a sphere and a plane $\pi$ . For a variable point $M \in \pi$ , exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone.

Octavian Stanasila

Let $OABC$ be a trihedral angle such that $$\angle BOC = \alpha, \angle COA = \beta, \angle AOB = \gamma , \alpha + \beta + \gamma = \pi .$$ For any interior point $P$ of the trihedral angle let $P_1$ , $P_2$ and $P_3$ be the projections of $P$ on the three faces. Prove that $OP \geq PP_1+PP_2+PP_3$ .

Constantin Cocea

Consider all regular convex and star polygons inscribed in a given circle and having $n$ sides. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist?

Mircea Becheanu

Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: $$(a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 .$$ Viorel Vajaitu

The cells of a $11\times 11$ chess-board are colored in 3 colors. Prove that there exists on the board a $m\times n$ rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color.

Ioan Tomescu

Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that $$\left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \ & \cdots & \ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right.$$ Mircea Becheanu

In the plane there are given the lines $\ell_1$ , $\ell_2$ , the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$ , $\ell_2$ and $\mathcal{C}$ . Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$ .

Mircea Becheanu

The positive integer $n$ is given and for all positive integers $k$ , $1\leq k\leq n$ , denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions:

a) $1\leq i_1<i_2< \cdots i_k \leq n$ ;

b) $i_{r+1}-i_r \equiv 1 \pmod 2$ , for all $r \in\{1,2,\ldots,k-1\}$ .

Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$ .

Ioan Tomescu

Let $p > 2$ be a prime number. Find the least positive number $a$ which can be represented as $$a = (X - 1)f(X) + (X^{p - 1} + X^{p - 2} + \cdots + X + 1)g(X),$$ where $f(X)$ and $g(X)$ are integer polynomials.

Mircea Becheanu.

Let $x,y,z$ be real numbers with $x+y+z=0$ . Prove that $$|\cos x |+ |\cos y| +| \cos z | \geq 1 .$$ Viorel Vajaitu, Bogdan Enescu

The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.

Laurentiu Panaitopol

Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$ , $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$ . Prove that $(y,x)$ is a solution of the equation $$|y^2 - axy - x^2 | = 1$$ if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$ .

Serban Buzeteanu

Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$ , for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$ . Find the set of values of $f$ .

Let $[a,b]$ be a given interval of real numbers not containing integers. Prove that there exists $N>0$ such that $[Na,Nb]$ does not contain integer numbers and the length of the interval $[Na,Nb]$ exceedes $\dfrac 16$ .

The finite sets $A_1$ , $A_2$ , $\ldots$ , $A_n$ are given and we denote by $d(n)$ the number of elements which appear exactly in an odd number of sets chosen from $A_1$ , $A_2$ , $\ldots$ , $A_n$ . Prove that for any $k$ , $1\leq k\leq n$ the number $${ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}|$$ is divisible by $2^k$ .

Ioan Tomescu, Dragos Popescu