41st International Mathematical Olympiad Taejon, Republic of Korea

Day I July 19, 2000

1. Two circles Γ_{1} and Γ_{2} intersect at M and N.

Let l be the common tangent to Γ_{1} and Γ_{2} so that M is closer to l than N is. Let l touch Γ_{1} at A and Γ_{2} at B. Let the line through M parallel to l meet the circle Γ_{1} again at C and the circle Γ_{2} again at D.

Lines CA and DB meet at E; lines AN and CD meet at P; lines BN and CD meet at Q.

Show that EP = EQ.

2. Let a, b, c be positive real numbers such that abc = 1. Prove that

(a − 1 + \frac{1}{b}) (b − 1 + \frac{1}{c}) (c − 1 + \frac{1}{a}) ≤ 1.

3. Let n ≥ 2 be a positive integer. Initially, there are n fleas on a horizontal line, not all at the same point.

For a positive real number λ, define a move as follows:

choose any two fleas, at points A and B, with A to the left of B; let the flea at A jump to the point C on the line to the right of B with BC/AB = λ.

Determine all values of λ such that, for any point M on the line and any initial positions of the n fleas, there is a finite sequence of moves that will take all the fleas to positions to the right of M.

Day II July 20, 2000

4. A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.

A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

5. Determine whether or not there exists a positive integer n such that

n is divisible by exactly 2000 different prime numbers, and 2^{n} + 1 is divisible by n.

6. Let AH_{1}, BH_{2}, CH_{3} be the altitudes of an acute-angled triangle ABC. The incircle of the triangle ABC touches the sides BC, CA, AB at T_{1}, T_{2}, T_{3}, respectively. Let the lines l_{1}, l_{2}, l_{3} be the reflections of the lines H_{2}H_{3}, H_{3}H_{1}, H_{1}H_{2} in the lines T_{2}T_{3}, T_{3}T_{1}, T_{1}T_{2}, respectively.

Prove that l_{1}, l_{2}, l_{3} determine a triangle whose vertices lie on the incircle of the triangle ABC.