TUYMAADA 2014

Given are three different primes. What maximum number of these primes can divide their sum?

(A. Golovanov)

For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$ . How many such linear functions $\ell(x)$ can exist?

(A. Golovanov)

Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$ . The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.

(L. Emelyanov)

There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$ . Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.

(K. Kokhas)

Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?

(A. Golovanov)

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that $$a_n-a_{n-1}+a_{n-2}- \cdots \ge 0$$ for each positive integer $n$ . Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$ .

(A. Golovanov)

A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$ . The line $DK$ meets the diagonal $AC$ at point $X$ ; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$ . Prove that the line $YL$ , median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point.

(A. Golovanov)

Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$ . Prove that $$g(a, b, c)\ge \sqrt{2abc}$$

(M. Ivanov)