2006 Turkey Team Selection Test

Part of Turkey Team Selection Test

Subcontests

(3)

2

xy+yz+zx=1 implies inequality

x,y,z

are positive real numbers and

xy+yz+zx=1

\frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3.

2006 students make a list with 12 schools

Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list. a) Prove that we can always find a good list with 12 elements, whatever the lists are; b) Prove that students can make lists such that no shorter list is good.

2

Ways to divide a rectangle in integer rectangles

How many ways are there to divide a

2\times n

rectangle into rectangles having integral sides, where

n

is a positive integer?

Prove that CD bisects QH

Q

on a circle with diameter

AB

A

B

, we draw a perpendicular to

AB

QH

H

AB

. The intersection points of the circle of diameter

AB

and the circle of center

Q

QH

C

D

CD

QH

2

Maximum area of a heptagon

Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.

Find least integer such that 2006 divides x_{2006}

For all integers

n\geq 1

x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2

x_1

is a positive integer. Find the least

x_1

such that 2006 divides

x_{2006}