2

Part of 1999 IMO Shortlist

Problems(4)

IMO ShortList 1999, geometry problem 2

Source: IMO ShortList 1999, geometry problem 2

A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

geometrypoint setcirclescombinatorial geometryIMO Shortlist

IMO ShortList 1999, number theory problem 2

Source: IMO ShortList 1999, number theory problem 2

Prove that every positive rational number can be represented in the form

\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}

where a,b,c,d are positive integers.

rationumber theoryrepresentationIMO Shortlist

IMO ShortList 1999, algebra problem 2

Source: IMO ShortList 1999, algebra problem 2

The numbers from 1 to

n^2

are randomly arranged in the cells of a

n \times n

n \geq 2

). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these

n^2\left(n-1\right)

fractions. What is the highest possible value of the characteristic ?

inequalitiescombinatoricsExtremal combinatoricsmatrixIMO Shortlist

IMO ShortList 1999, combinatorics problem 2

Source: IMO ShortList 1999, combinatorics problem 2

5 \times n

rectangle can be tiled using

n

pieces like those shown in the diagram, prove that

n

is even. Show that there are more than

2 \cdot 3^{k-1}

ways to file a fixed

5 \times 2k

(k \geq 3)

2k

pieces. (symmetric constructions are supposed to be different.)

geometryrectanglecombinatoricscountingIMO Shortlist