2010 International Zhautykov Olympiad

Part of International Zhautykov Olympiad

Subcontests

(3)

2

IZhO 2010 1-st round-2

In a cyclic quadrilateral

ABCD

AB=AD

M

N

lie on the sides

BC

CD

respectively so that

MN=BM+DN

AM

AN

meet the circumcircle of

ABCD

again at points

P

Q

respectively. Prove that the orthocenter of the triangle

APQ

lies on the segment

MN

IZhO 2010 2-nd round-2

In every vertex of a regular

n

-gon exactly one chip is placed. At each

step

one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by

[\frac{n}{2}]

positions clockwise from its initial position.

2

CNT from IZhO 2010 2-nd round-1

Positive integers

1,2,...,n

are written on а blackboard (

n >2

). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least

n

for which it is possible.

IZhO 2010 1-st round-1

Find all primes

p,q

p^3-q^7=p-q

2

Problem by IZhO 2010

ABC

arbitrary triangle (

AB \neq BC \neq AC \neq AB

) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1)

\angle OIH > 90^0

\angle OIH >135^0

(7 points)
balls for 1) and 2) not additive.

IZhO 2010 1-st round-3

A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
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