MathDB

a convex hexagon "inscribed" in a polygon with area 3/4

Source: mock tst romania 2004

12/31/2004

Let

P

be a convex polygon. Prove that there exists a convex hexagon that is contained in

P

and whose area is at least

\frac34

of the area of the polygon

P

.
Alternative version. Let

P

be a convex polygon with

n\geq 6

vertices. Prove that there exists a convex hexagon with
a) vertices on the sides of the polygon (or) b) vertices among the vertices of the polygon
such that the area of the hexagon is at least

\frac{3}{4}

of the area of the polygon.
Proposed by Ben Green and Edward Crane, United Kingdom

geometryareapolygonconvex polygongeometric inequalityIMO Shortlist

P_{n} the product of all positive integers x less than n

Source: IMO Shortlist 2004, number theory problem 6

6/14/2005

Given an integer

{n>1}

, denote by

P_{n}

the product of all positive integers

x

less than

n

and such that

n

divides

{x^2-1}

. For each

{n>1}

, find the remainder of

P_{n}

on division by

n

.
Proposed by John Murray, Ireland

modular arithmeticnumber theoryDivisibilityremainderIMO Shortlist

shortlisted - function equation

Source: IMO ShortList 2004, algebra problem 6

4/16/2005

Find all functions

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

satisfying the equation

f(x^2+y^2+2f(xy)) = (f(x+y))^2.

for all

x,y \in \mathbb{R}

.

functioncalculusalgebrafunctional equationIMO Shortlist

Good Problem.

Source: IMO ShortList 2004, combinatorics problem 6

3/23/2005

For an

{n\times n}

matrix

A

, let

X_{i}

be the set of entries in row

i

, and

Y_{j}

the set of entries in column

j

,

{1\leq i,j\leq n}

. We say that

A

is golden if

{X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}

are distinct sets. Find the least integer

n

such that there exists a

{2004\times 2004}

golden matrix with entries in the set

{\{1,2,\dots ,n\}}

.

linear algebramatrixcombinatoricsExtremal combinatoricsIMO Shortlist

6

Problems(4)