1998 JBMO Problems

Problem 1

Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon.

Find all pairs of positive integers $(x,y)$ such that $x^y = y^{x - y}.$

Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16?

1998 JBMO (Problems • Resources)
Preceded by 1997 JBMO Problems	Followed by 1999 JBMO Problems
1 • 2 • 3 • 4
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