5

Part of 2008 IMO Shortlist

Problems(4)

IMO ShortList 2008, Combinatorics problem 5

Source: IMO ShortList 2008, Combinatorics problem 5, German TST 2, P3, 2009

Let S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\} be a (k \plus{} l)-element set of real numbers contained in the interval

[0, 1]

k

l

are positive integers. A

k

-element subset

A\subset S

is called nice if \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl} Prove that the number of nice subsets is at least \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}. Proposed by Andrey Badzyan, Russia

combinatoricsProbabilistic MethodcountingIMO Shortlist

IMO ShortList 2008, Algebra problem 5

Source: IMO ShortList 2008, Algebra problem 5, German TST 1, P3, 2009

a

b

c

d

be positive real numbers such that abcd \equal{} 1 and a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}. Prove that a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d} Proposed by Pavel Novotný, Slovakia

inequalitiesalgebraIMO Shortlist

IMO Shortlist 2008, Geometry problem 5

Source: IMO Shortlist 2008, Geometry problem 5, German TST 1, P2, 2009

k

n

be integers with 0\le k\le n \minus{} 2. Consider a set

L

n

lines in the plane such that no two of them are parallel and no three have a common point. Denote by

I

the set of intersections of lines in

L

O

be a point in the plane not lying on any line of

L

X\in I

is colored red if the open line segment

OX

intersects at most

k

L

I

contains at least \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2) red points. Proposed by Gerhard Woeginger, Netherlands

geometrypoint setcombinatorial geometrylinesIMO Shortlist

IMO ShortList 2008, Number Theory problem 5

Source: IMO ShortList 2008, Number Theory problem 5, German TST 6, P2, 2009

n\in\mathbb{N}

d(n)

denote the number of (positive) divisors of

n

. Find all functions

f: \mathbb{N}\to\mathbb{N}

with the following properties: [*] d\left(f(x)\right) \equal{} x for all

x\in\mathbb{N}

f(xy)

divides (x \minus{} 1)y^{xy \minus{} 1}f(x) for all

x

y\in\mathbb{N}

. Proposed by Bruno Le Floch, France

functionnumber theorymodular arithmeticdivisorIMO Shortlistfunctional equation