2000 IMO Problems

DAY 1

Problem 1

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP=EQ$ .

Solution

Problem 2

Let $a, b, c$ be positive real numbers with $abc=1$ . Show that

$\left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1$

Solution

Problem 3

Let $n \ge 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$ to the left of $B$ , and letting the flea from $A$ jump over the flea from $B$ to the point $C$ so that $\frac{BC}{AB}=\lambda$ .

Determine all values of $\lambda$ such that, for any point $M$ on the line and for any initial position of the $n$ fleas, there exists a sequence of moves that will take them all to the position right of $M$ .

Solution

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2000 IMO Problems

Contents

DAY 1

Problem 1

Problem 2

Problem 3