2012 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

(4)

4

4

3

Easy polynomial problem

P(x)

be a real quadratic trinomial, so that for all

x\in \mathbb{R}

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

Both cyclic and circumscribed quadrilateral

ABCD

is both cyclic and circumscribed. Its incircle touches its sides

AB

CD

X

Y

, respectively. The perpendiculars to

AB

CD

A

D

, respectively, meet at point

U

; those drawn at

X

Y

V

, and finally, those drawn at

B

C

W

. Prove that points

U

V

W

are collinear.
Proposed by A. Golovanov

Angle bisector in a rectangle

ABCD

is given. Segment

DK

BD

and lies on the half-line

DC

M

is the midpoint of

BK

AM

is the angle bisector of

\angle BAC

.
Proposed by S. Berlov

3

Chips on a chessboard

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
Proposed by A. Golovanov

Diophantine equation

Solve in positive integers the following equation:

{1\over n^2}-{3\over 2n^3}={1\over m^2}

Proposed by A. Golovanov

Labeling of the vertices of a regular 2012-gon

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

m+n

leave the same remainder when divided by

2012

, then the chords

A_kA_{\ell}

A_mA_n

have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled

A_1

A_4

. How is the tenth vertex labeled?
Proposed by A. Golovanov