MathDB

5

Part of 2007 IMO Shortlist

Problems(4)

a(m+n) >= 2 a(m) + 2 a(n) for all m,n >= 1

Source: IMO Shortlist 2007, A5

7/13/2008
Let c>2, c > 2, and let a(1),a(2),… a(1), a(2), \ldots be a sequence of nonnegative real numbers such that a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1, and a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0. Prove that the sequence a(n) a(n) is bounded. Author: Vjekoslav Kovač, Croatia
inequalitiesSequenceboundedrecurrence relationIMO Shortlist
Placing a rectangle which is not a square in a painted plane

Source: Contest "Scoala Cu Ceas" 2008 Seniors Problem 3 (Day 1), approx. IMO Shortlist 2007 Problem C5

3/20/2008
In the Cartesian coordinate plane define the strips S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}, n∈Z n\in\mathbb{Z} and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.
IMO Shortlist 2007 Problem C5 as it appears in the official booklet: In the Cartesian coordinate plane define the strips S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\} for every integer n. n. Assume each strip Sn S_n is colored either red or blue, and let a a and b b be two distinct positive integers. Prove that there exists a rectangle with side length a a and b b such that its vertices have the same color. (Edited by Orlando Döhring)
Author: Radu Gologan and Dan Schwarz, Romania
geometryrectanglecombinatoricsRamsey TheoryColoringIMO ShortlistRIP mavropnevma
Problem G5 - IMO Shortlist 2007

Source: ISL 2007, G5, AIMO 2008, TST 3, P2

7/13/2008
Let ABC ABC be a fixed triangle, and let A1 A_1, B1 B_1, C1 C_1 be the midpoints of sides BC BC, CA CA, AB AB, respectively. Let P P be a variable point on the circumcircle. Let lines PA1 PA_1, PB1 PB_1, PC1 PC_1 meet the circumcircle again at A′ A', B′ B', C′ C', respectively. Assume that the points A A, B B, C C, A′ A', B′ B', C′ C' are distinct, and lines AA′ AA', BB′ BB', CC′ CC' form a triangle. Prove that the area of this triangle does not depend on P P. Author: Christopher Bradley, United Kingdom
geometrycircumcircleIMO Shortlist
$p|f(m+n) \iff p|f(m) + f(n)$ (IMO Shortlist 2007, N5)

Source: IMO Shortlist 2007, N5, AIMO 2008, TST 3, P3

7/13/2008
Find all surjective functions f:N→N f: \mathbb{N} \to \mathbb{N} such that for every m,n∈N m,n \in \mathbb{N} and every prime p, p, the number f(m+n) f(m + n) is divisible by p p if and only if f(m)+f(n) f(m) + f(n) is divisible by p p.
Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran
functionmodular arithmeticnumber theoryDivisibilityIMO Shortlist