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Competition Problems

IMO 2000 Problems

Taejon, Korea
July 13-25, 2000

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Problem 1

AB 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 2

A, B, C are positive reals with product 1. Prove that (A - 1 + 1/B) (B - 1 + 1/C) (C - 1 + 1/A) <= 1.


Problem 3

k 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 4

100 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 5

Can we find N divisible by just 2000 different primes, so that N divides 2N + 1? [N may be divisible by a prime power.]


Problem 6

A1 A2 A3 is an acute-angled triangle. The foot of the altitude from Ai is Ki and the incircle touches the side opposite Ai at Li. The line K1 K2 is reflected in the line L1 L2. Similarly, the line K2 K3 is reflected in L2 L3 and K3 K1 is reflected in L3 L1. Show that the three new lines form a triangle with vertices on the incircle.