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.


Problem 2

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.


Problem 3

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


Problem 4

Do[es] 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?


See Also

1998 JBMO (ProblemsResources)
Preceded by
1997 JBMO Problems
Followed by
1999 JBMO Problems
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All JBMO Problems and Solutions