TUYMAADA 2005

The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers?

Proposed by S. Berlov

Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$ .

The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$ .

For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others?

Proposed by F. Petrov, K. Sukhov

The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.

Proposed by S. Berlov, S. Ivanov

In a triangle $ABC$ , let $A_{1}$ , $B_{1}$ , $C_{1}$ be the points where the excircles touch the sides $BC$ , $CA$ and $AB$ respectively. Prove that $A A_{1}$ , $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.

Proposed by L. Emelyanov

You have $2$ columns of $11$ squares in the middle, in the right and in the left you have columns of $9$ squares (centered on the ones of $11$ squares), then columns of $7,5,3,1$ squares. (This is the way it was explained in the original thread, http://www.artofproblemsolving.com/Forum/viewtopic.php?t=44430 ; anyway, i think you can understand how it looks)

Several rooks stand on the table and beat all the squares ( a rook beats the square it stands in, too). Prove that one can remove several rooks such that not more than $11$ rooks are left and still beat all the table.

Proposed by D. Rostovsky, based on folklore

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$ , $\ldots$ , $x_i=\frac{a_i}{n+i}$ , with $a_i < n+i$ . Prove that there exist a positive integer $k$ and integers $c_1$ , $\ldots$ , $c_k$ such that $$c_1 x_1 + \ldots + c_k x_k = 1.$$

Proposed by M. Dubashinsky

Let $I$ be the incentre of triangle $ABC$ . A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$ . Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$ .

Proposed by S. Berlov

Let $a,b,c$ be positive reals s.t. $a^2+b^2+c^2=1$ . Prove the following inequality $$\sum \frac{a}{a^3+bc} >3 .$$

Proposed by A. Khrabrov