1980 USAMO Problems
Problems from the 1980 USAMO.
A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight , when placed in the left pan and against a weight , when placed in the right pan. The corresponding weights for the second object are and . The third object balances against a weight , when placed in the left pan. What is its true weight?
Find the maximum possible number of three term arithmetic progressions in a monotone sequence of distinct reals.
is an integral multiple of . and are real numbers. If , show that for any positive integer .
The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
If are reals such that , show that
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