1975 USAMO Problems
Problems from the 1975 USAMO.
(a) Prove that
where and denotes the greatest integer (e.g., ).
(b) Using (a) or otherwise, prove that
is integral for all positive integral and .
Let denote four points in space and the distance between and , and so on. Show that
If denotes a polynomial of degree such that for , determine .
Two given circles intersect in two points and . Show how to construct a segment passing through and terminating on the two circles such that is a maximum.
A deck of playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is .
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