2009 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(3)

2

roots of a cubic

Denote the three real roots of the cubic x^3 \minus{} 3x \minus{} 1 \equal{} 0 by

x_1

x_2

x_3

in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1.

Inequality 3 variables - seems familiar

x

y

z

be non negative numbers. Prove that \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}

2

Does there exist a pair of strictly monotonic funct

Does there exist a pair

(f; g)

of strictly monotonic functions, both from

\mathbb{N}

\mathbb{N}

f(g(g(n))) < g(f(n))

n \in\mathbb{N}

Do there exist 2009 distinct positive integers

(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers? (b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?

2

Find the values of k

For a given prime

p > 2

and positive integer

k

let S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k Find those values of

k

p \, |\, S_k

Prove that there exists a point Q

Given is the convex quadrilateral

ABCD

. Assume that there exists a point

P

inside the quadrilateral for which the triangles

ABP

CDP

are both isosceles right triangles with the right angle at the common vertex

P

. Prove that there exists a point

Q

for which the triangles

BCQ

ADQ

are also isosceles right triangles with the right angle at the common vertex

Q