RO-TST 1996

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that for every regular $n$ -gon $A_1A_2 \ldots A_n$ we have $f(A_1)+f(A_2)+\cdots +f(A_n)=0$ . Prove that $f(x)=0$ for all reals $x$ .

Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$ , not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$ , not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$ .

Let $x,y\in \mathbb{R}$ . Show that if the set $A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\}$ is finite then $x,y \in \mathbb{Q}$ .

Vasile Pop

Let $ABCD$ be a cyclic quadrilateral and let $M$ be the set of incenters and excenters of the triangles $BCD$ , $CDA$ , $DAB$ , $ABC$ (so 16 points in total). Prove that there exist two sets $\mathcal{K}$ and $\mathcal{L}$ of four parallel lines each, such that every line in $\mathcal{K} \cup \mathcal{L}$ contains exactly four points of $M$ .

Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$ . Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$ , $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$ . Evaluate the value of $\angle PSR$ .

Let $a\in \mathbb{R}$ and $f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R}$ are the additive functions such that for every $x\in \mathbb{R}$ we have $f_1(x)f_2(x) \cdots f_n(x) =ax^n$ . Show that there exists $b\in \mathbb {R}$ and $i\in {\{1,2,\ldots,n}\}$ such that for every $x\in \mathbb{R}$ we have $f_i(x)=bx$ .

Let $p_1,p_2,\ldots,p_k$ be the distinct prime divisors of $n$ and let $a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k}$ for $n\geq 2$ . Show that for every positive integer $N\geq 2$ the following inequality holds: $\sum_{k=2}^{N} a_2a_3 \cdots a_k <1$

Laurentiu Panaitopol

Let $n\geq 3$ be an integer and let $x_1,x_2,\ldots,x_{n-1}$ be nonnegative integers such that

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of $F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k$ .

Let $n$ and $r$ be positive integers and $A$ be a set of lattice points in the plane such that any open disc of radius $r$ contains a point of $A$ . Show that for any coloring of the points of $A$ in $n$ colors there exists four points of the same color which are the vertices of a rectangle.

Find all primes $p,q$ such that $\alpha^{3pq} -\alpha \equiv 0 \pmod {3pq}$ for all integers $\alpha$ .

Let $n\geq 3$ be an integer and let $p\geq 2n-3$ be a prime number. For a set $M$ of $n$ points in the plane, no 3 collinear, let $f: M\to \{0,1,\ldots, p-1\}$ be a function such that

(i) exactly one point of $M$ maps to 0, (ii) if a circle $\mathcal{C}$ passes through 3 distinct points of $A,B,C\in M$ then $\sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p$ .

Prove that all the points in $M$ lie on a circle.

Let $x_1,x_2,\ldots,x_n$ be positive real numbers and $x_{n+1} = x_1 + x_2 + \cdots + x_n$ . Prove that $$\sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}.$$ Mircea Becheanu

Let $x,y,z$ be real numbers. Prove that the following conditions are equivalent:

(i) $x,y,z$ are positive numbers and $\dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1$ ; (ii) $a^2x+b^2y+c^2z>d^2$ holds for every quadrilateral with sides $a,b,c,d$ .

Let $S$ be a set of $n$ concentric circles in the plane. Prove that if a function $f: S\to S$ satisfies the property $$d( f(A),f(B)) \geq d(A,B)$$ for all $A,B \in S$ , then $d(f(A),f(B)) = d(A,B)$ , where $d$ is the euclidean distance function.

Let $n\geq 3$ be an integer and let $\mathcal{S} \subset \{1,2,\ldots, n^3\}$ be a set with $3n^2$ elements. Prove that there exist nine distinct numbers $a_1,a_2,\ldots,a_9 \in \mathcal{S}$ such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 a_4x + a_5 y + a_6 z &=& 0 a_7x + a_8y + a_9z &=& 0. \end{eqnarray*}

Marius Cavachi