2

Part of 1996 IMO Shortlist

Problems(2)

(x - a_1)(x - a_2) ... (x - a_n) <= x^n - a^n_1

Source: IMO Shortlist 1996, A2

a_1 \geq a_2 \geq \ldots \geq a_n

be real numbers such that for all integers

k > 0,

a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.
Let p \equal{}\max\{|a_1|, \ldots, |a_n|\}. Prove that p \equal{} a_1 and that
(x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1 for all

x > a_1.

algebraSequenceInequalityIMO Shortlist

color the squares

Source: IMO Shortlist 1996, C2, Hungarian MO 2000, Italian MO 2008/2

A square (n \minus{} 1) \times (n \minus{} 1) is divided into (n \minus{} 1)^2 unit squares in the usual manner. Each of the

n^2

vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

geometryrectanglecombinatoricsColoringcountingIMO Shortlist