2000 USAMO Problems
Call a real-valued function very convex if
holds for all real numbers and . Prove that no very convex function exists.
Let be the set of all triangles for which
where is the inradius and are the points of tangency of the incircle with sides respectively. Prove that all triangles in are isosceles and similar to one another.
A game of solitaire is played with red cards, white cards, and blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of and the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Find the smallest positive integer such that if squares of a chessboard are colored, then there will exist three colored squares whose centers form a right triangle with legs parallel to the edges of the board.
Let be a triangle and let be a circle in its plane passing through and Suppose there exist circles such that for is externally tangent to and passes through and where for all . Prove that
Let be nonnegative real numbers. Prove that
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