MathDB

2

Part of 2002 IMO Shortlist

Problems(3)

IMO ShortList 2002, geometry problem 2

Source: IMO ShortList 2002, geometry problem 2

9/28/2004
Let ABCABC be a triangle for which there exists an interior point FF such that ∠AFB=∠BFC=∠CFA\angle AFB=\angle BFC=\angle CFA. Let the lines BFBF and CFCF meet the sides ACAC and ABAB at DD and EE respectively. Prove that AB+AC≥4DE. AB+AC\geq4DE.
geometryhomothetyinequalitiesgeometric inequalityTriangleIMO Shortlist
IMO ShortList 2002, algebra problem 2

Source: IMO ShortList 2002, algebra problem 2

9/28/2004
Let a1,a2,…a_1,a_2,\ldots be an infinite sequence of real numbers, for which there exists a real number cc with 0≤ai≤c0\leq a_i\leq c for all ii, such that \left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j}   \text{for all }i,\ j \text{ with } i \neq j. Prove that c≥1c\geq1.
inequalitiesalgebraSequenceboundedIMO Shortlist
IMO ShortList 2002, combinatorics problem 2

Source: IMO ShortList 2002, combinatorics problem 2

9/28/2004
For nn an odd positive integer, the unit squares of an n×nn\times n chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an LL-shape formed by three connected unit squares. For which values of nn is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?
combinatoricsTilingdissectionIMO Shortlist