1998 JBMO Problems

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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.

Solution

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.

Solution

Problem 3

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

Solution

Problem 4

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?

Solution

See Also

1998 JBMO (ProblemsResources)
Preceded by
1997 JBMO Problems
Followed by
1999 JBMO Problems
1 2 3 4
All JBMO Problems and Solutions


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