International Mathematical Olympiad · 2000

Problems

Problem 1Two circles $\Gamma_1$ and $\Gamma_2$ intersect at two points $M$ and $N$ . Let $AB$ be the common tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let the line through $M$ parallel to $AB$ meet $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Prove that $EP=EQ$ .Solutions: 1
Problem 2Let $a,b,c$ be positive real numbers with $abc=1$ . Prove that $(a-1+\frac1b)(b-1+\frac1c)(c-1+\frac1a)\le 1$ .Solutions: 1
Problem 3Let $n\ge2$ be a positive integer. Initially there are $n$ fleas on a horizontal line, not all at the same point. For a positive real number $\lambda$ , a move chooses fleas at $A$ and $B$ with $A$ to the left of $B$ , and lets the flea at $A$ jump to $C$ to the right of $B$ so that $BC=\lambda AB$ . Determine all $\lambda$ such that, for every point $M$ and every initial position, a finite sequence of moves puts all fleas to the right of $M$ .Solutions: 1
Problem 4A magician has one hundred cards numbered $1$ to $100$ . He puts them into three boxes, red, white and blue, each nonempty. A member selects two boxes, chooses one card from each, and announces their sum. Given this sum, the magician identifies the box from which no card was chosen. How many assignments of the cards to the three boxes make this always possible?Solutions: 1
Problem 5Does there exist a positive integer $n$ such that $n$ has exactly $2000$ distinct prime divisors and $n$ divides $2^n+1$ ?Solutions: 1
Problem 6Let $AH_1,BH_2,CH_3$ be the altitudes of an acute triangle $ABC$ . The incircle of $ABC$ touches $BC,CA,AB$ at $T_1,T_2,T_3$ , respectively. Reflect the lines $H_1H_2,H_2H_3,H_3H_1$ in the lines $T_1T_2,T_2T_3,T_3T_1$ , respectively. Prove that the three reflected lines form a triangle whose vertices lie on the incircle.Solutions: 1