This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

1980 Putnam, A4
a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$ |a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $ a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$ |a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$
putnam pigeonhole pigeonhole principle inequalities

2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
putnam pigeonhole geometry 3d geometry pigeonhole principle sphere solid geometry