TUYMAADA 2012

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$ . $M$ is the midpoint of $BK$ . Prove that $AM$ is the angle bisector of $\angle BAC$ .

Proposed by S. Berlov

Prove that $N^2$ arbitrary distinct positive integers ( $N>10$ ) can be arranged in a $N\times N$ table, so that all $2N$ sums in rows and columns are distinct.

Proposed by S. Volchenkov

Let $p=1601$ . Prove that if $$\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},$$ where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$ .

Proposed by A. Golovanov

The vertices of a regular $2012$ -gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$ , then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$ . How is the tenth vertex labeled?

Proposed by A. Golovanov

A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$ . Prove that the length of its radius is less than $3$ .

Proposed by K. Kokhas

$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the $24$ remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?

Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.

Proposed by F. Petrov

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$ .

Proposed by A. Golovanov, M. Ivanov, K. Kokhas

Point $P$ is taken in the interior of the triangle $ABC$ , so that $$\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).$$ Let $L$ be the foot of the angle bisector of $\angle B$ . The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$ . Prove that $QB$ is the angle bisector of $\angle AQC$ .

Proposed by S. Berlov

Let $p=4k+3$ be a prime. Prove that if $$\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}$$ (where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$ .

Proposed by A. Golovanov

Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$ , respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$ , respectively, meet at point $U$ ; those drawn at $X$ and $Y$ meet at point $V$ , and finally, those drawn at $B$ and $C$ meet at point $W$ . Prove that points $U$ , $V$ and $W$ are collinear.

Proposed by A. Golovanov

Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds $$\dfrac{1} {2a^2+b^2+3}+\dfrac {1} {2b^2+c^2+3}+\dfrac{1} {2c^2+a^2+3}\leq \dfrac {1} {2}.$$

Proposed by V. Aksenov

Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the weight of a vertexthe difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$ . Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero.

Proposed by W. Tutte