MathDB

2p pairwise distinct subsets s.t. intersection non-empty

Source: IMO Shortlist 2007, C7

7/13/2008

Let \alpha < \frac {3 \minus{} \sqrt {5}}{2} be a positive real number. Prove that there exist positive integers

n

and

p > \alpha \cdot 2^n

for which one can select

2 \cdot p

pairwise distinct subsets

S_1, \ldots, S_p, T_1, \ldots, T_p

of the set

\{1,2, \ldots, n\}

such that

S_i \cap T_j \neq \emptyset

for all

1 \leq i,j \leq p

Author: Gerhard Wöginger, Austria

combinatoricsSet systemsExtremal combinatoricsIMO Shortlist

circumradius, incenter, ...

Source: IMO Shortlist 2007, G7

7/6/2008

Given an acute triangle

ABC

with

\angle B > \angle C

. Point

I

is the incenter, and

R

the circumradius. Point

D

is the foot of the altitude from vertex

A

. Point

K

lies on line

AD

such that AK \equal{} 2R, and

D

separates

A

and

K

. Lines

DI

and

KI

meet sides

AC

and

BC

at

E,F

respectively. Let IE \equal{} IF. Prove that

\angle B\leq 3\angle C

. Author: Davoud Vakili, Iran

geometrycircumcircleincenterreflectionIMO Shortlist

Prime factorisation of n!

Source: ISL 2007 N7

7/13/2008

For a prime

p

and a given integer

n

let

\nu_p(n)

denote the exponent of

p

in the prime factorisation of

n!

. Given

d \in \mathbb{N}

and

\{p_1,p_2,\ldots,p_k\}

a set of

k

primes, show that there are infinitely many positive integers

n

such that

d\mid \nu_{p_i}(n)

for all

1 \leq i \leq k

.
Author: Tejaswi Navilarekkallu, India

modular arithmeticnumber theoryprime factorizationfactorialIMO ShortlistCombinatorial Number Theory

7

Problems(3)