TUYMAADA 2009

All squares of a $20\times 20$ table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy?

Proposed by A. Golovanov

US Name Conversions:

Misha: Naoki Sasha**: Richard*

$P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $P(1)$ , $P(2)$ , $P(3)$ , $\dots?$

Proposed by A. Golovanov

In a cyclic quadrilateral $ABCD$ the sides $AB$ and $AD$ are equal, $CD>AB+BC$ . Prove that $\angle ABC>120^\circ$ .

Each of the subsets $A_1$ , $A_2$ , $\dots,$ $A_n$ of a 2009-element set $X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $X$ there is a 24-element subset $B$ containing neither of the sets $A_1$ , $A_2$ , $\dots,$ $A_n$ .

$M$ is the midpoint of base $BC$ in a trapezoid $ABCD$ . A point $P$ is chosen on the base $AD$ . The line $PM$ meets the line $CD$ at a point $Q$ such that $C$ lies between $Q$ and $D$ . The perpendicular to the bases drawn through $P$ meets the line $BQ$ at $K$ . Prove that $\angle QBC = \angle KDA$ .

Proposed by S. Berlov

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.

Proposed by A. Khabrov

Three real numbers are given. Fractional part of the product of every two of them is $1\over 2$ . Prove that these numbers are irrational.

Proposed by A. Golovanov

A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?

Proposed by A. Golovanov

On the side $AB$ of a cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $BD$ bisects $CX$ and diagonal $AC$ bisects $DX$ . What is the minimum possible value of $AB\over CD$ ?

Proposed by S. Berlov

Is there a positive integer $n$ such that among 200th digits after decimal point in the decimal representations of $\sqrt{n}$ , $\sqrt{n+1}$ , $\sqrt{n+2}$ , $\ldots,$ $\sqrt{n+999}$ every digit occurs 100 times?

Proposed by A. Golovanov

A magician asked a spectator to think of a three-digit number $\overline{abc}$ and then to tell him the sum of numbers $\overline{acb}$ , $\overline{bac}$ , $\overline{bca}$ , $\overline{cab}$ , and $\overline{cba}$ . He claims that when he knows this sum he can determine the original number. Is that so?

An arrangement of chips in the squares of $n\times n$ table is called sparse if every $2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $n$ is this possible?

Proposed by S. Berlov

A triangle $ABC$ is given. Let $B_1$ be the reflection of $B$ across the line $AC$ , $C_1$ the reflection of $C$ across the line $AB$ , and $O_1$ the reflection of the circumcentre of $ABC$ across the line $BC$ . Prove that the circumcentre of $AB_1C_1$ lies on the line $AO_1$ .

Proposed by A. Akopyan

Determine the maximum number $h$ satisfying the following condition: for every $a\in [0,h]$ and every polynomial $P(x)$ of degree 99 such that $P(0)=P(1)=0$ , there exist $x_1,x_2\in [0,1]$ such that $P(x_1)=P(x_2)$ and $x_2-x_1=a$ .

Proposed by F. Petrov, D. Rostovsky, A. Khrabrov