2013 Benelux

Part of Benelux

Subcontests

(4)

1

Integers g for which all primes divide some g^n-n

a) Find all positive integers

g

with the following property: for each odd prime number

p

there exists a positive integer

n

p

divides the two integers g^n - n \text{ and } g^{n+1} - (n + 1). b) Find all positive integers

g

with the following property: for each odd prime number

p

there exists a positive integer

n

p

divides the two integers g^n - n^2 \text{ and }g^{n+1} - (n + 1)^2.

1

Triple product of three sides equals another triple product

\triangle ABC

be a triangle with circumcircle

\Gamma

I

be the center of the incircle of

\triangle ABC

AI

BI

CI

\Gamma

D \ne A

E \ne B

F \ne C

. The tangent lines to

\Gamma

F

D

E

intersect the lines

AI

BI

CI

R

S

T

, respectively. Prove that

\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.

1

Functional: f(x+y)+y<f^(3)(x)

Find all functions

f: \mathbb{R} \to \mathbb{R}

f(x + y) + y \le f(f(f(x)))

x, y \in \mathbb{R}

1

How many points can the frog miss?

n \ge 3

be an integer. A frog is to jump along the real axis, starting at the point

0

n

jumps: one of length

1

, one of length

2

\dots

, one of length

n

. It may perform these

n

jumps in any order. If at some point the frog is sitting on a number

a \le 0

, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number

a > 0

, its next jump must be to the left (towards the negative numbers). Find the largest positive integer

k

for which the frog can perform its jumps in such an order that it never lands on any of the numbers

1, 2, \dots , k