RO-TST 2004

Let $a_1,a_2,a_3,a_4$ be the sides of an arbitrary quadrilateral of perimeter $2s$ . Prove that $$\sum\limits^4_{i=1} \dfrac 1{a_i+s} \leq \dfrac 29\sum\limits_{1\leq i<j\leq 4} \dfrac 1{ \sqrt { (s-a_i)(s-a_j)}}.$$ When does the equality hold?

Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\bigcup_{i=1}^n R_i$ which has an area of at least 1.

[Thanks Grobber for the correction]

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$ , and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$ .

Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$ , $x_1=0$ , $x_2=2c$ , $x_3=3b$ and for all positive integers $n$ we have $$x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} .$$ Prove that for all positive integers $m$ , and for all primes $p$ the number $x_{p^m}$ is divisible by $p$ .

Let $\Gamma$ be a circle, and let $ABCD$ be a square lying inside the circle $\Gamma$ . Let $\mathcal{C}_a$ be a circle tangent interiorly to $\Gamma$ , and also tangent to the sides $AB$ and $AD$ of the square, and also lying inside the opposite angle of $\angle BAD$ . Let $A'$ be the tangency point of the two circles. Define similarly the circles $\mathcal{C}_b$ , $\mathcal{C}_c$ , $\mathcal{C}_d$ and the points $B',C',D'$ respectively.

Prove that the lines $AA'$ , $BB'$ , $CC'$ and $DD'$ are concurrent.

Attachments:

Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.

Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

Prove that for all positive integers $n,m$ , with $m$ odd, the following number is an integer $$\frac 1{3^mn}\sum^m_{k=0} { 3m \choose 3k } (3n-1)^k.$$

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$ , the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$ . Prove that the lines $IP$ and $MN$ are perpendicular.

Alternative formulation. The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$ , $CA$ and $AB$ in $A^{\prime}$ , $B^{\prime}$ and $C^{\prime}$ , respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$ , the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$ , and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$ . Prove that the lines $IP$ and $MN$ are perpendicular.

Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have $$\left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 .$$ Gabriel Dospinescu

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$ , we have $a^m - 1\equiv 0 \pmod n$ .

Prove that $n \leq 4m(2^m-1)$ .

Created by Harazi, modified by Marian Andronache.

Let $O$ be a point in the plane of the triangle $ABC$ . A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$ , $COA$ and $AOB$ respectively in $K,L,M$ .

Prove that the lines $PK,QL$ and $RM$ are concurrent.

Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white.

Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$ .

Three circles $\mathcal{K}_1$ , $\mathcal{K}_2$ , $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$ . The point $O$ lies inside the triangle $ABC$ .

Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$ . Let $\alpha = \frac {OA_1}{AA_1}$ , $\beta= \frac {OB_1}{BB_1}$ and $\gamma = \frac {OC_1}{CC_1}$ and let $R$ be the circumradius of the triangle $ABC$ . Prove that $$\alpha R_1 + \beta R_2 + \gamma R_3 \geq R.$$

On a chess table $n\times m$ we call a move the following succesion of operations

(i) choosing some unmarked squares, any two not lying on the same row or column; (ii) marking them with 1; (iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move).

We call a gamea succession of moves that end in the moment that we cannot make any more moves.

What is the maximum possible sum of the numbers on the table at the end of a game?

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by $$f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 ,$$ where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$ ).

a) Prove that $f(x)$ is divisible with $(x-1)$ , but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$ ; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$ .

Sugested by Calin Popescu.