1

Part of 2009 Hungary-Israel Binational

Problems(2)

Find the values of k

Source: Hungary-Israel Binational Olympiad 2009, Problem 1

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

modular arithmeticalgebrapolynomialnumber theoryrelatively primenumber theory unsolved

Prove that there exists a point Q

Source: Hungary-Israel Binational Olympiad 2009, Problem 4

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

geometrytrapezoidcircumcircleparallelogrampower of a pointradical axiscomplex numbers