IMO 2000 ProblemsTaejon, KoreaJuly 13-25, 2000 Download the problems as: Mathematica Notebook PDF Problem 1AB is tangent to the circles CAMN and NMBD. M lies between C and D on the line CD, and CD is parallel to AB. The chords NA and CM meet at P; the chords NB and MD meet at Q. The rays CA and DB meet at E. Prove that PE = QE. Problem 2A, B, C are positive reals with product 1. Prove that (A - 1 + 1/B) (B - 1 + 1/C) (C - 1 + 1/A) <= 1. Problem 3k is a positive real. N is an integer greater than 1. N points are placed on a line, not all coincident. A move is carried out as follows. Pick any two points A and B which are not coincident. Suppose that A lies to the right of B. Replace B by another point B' to the right of A such that AB' = kBA. For what values of k can we move the points arbitrarily far to the right by repeated moves?Problem 4100 cards are numbered 1 to 100 (each card different) and placed in three boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box? Problem 5Can we find N divisible by just 2000 different primes, so that N divides 2^{N} + 1? [N may be divisible by a prime power.] Problem 6A_{1} A_{2} A_{3} is an acute-angled triangle. The foot of the altitude from A_{i} is K_{i} and the incircle touches the side opposite A_{i} at L_{i}. The line K_{1} K_{2} is reflected in the line L_{1} L_{2}. Similarly, the line K_{2} K_{3} is reflected in L_{2} L_{3} and K_{3} K_{1} is reflected in L_{3} L_{1}. Show that the three new lines form a triangle with vertices on the incircle. |