Problem 12 (2015 AIME I)
AIME 2015
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Understand the scale
Scale
0.5: Easiest math competition problems, often solvable even for early middle school students without experience.
1: Problems strictly for beginners, often at a moderate middle school level.
1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests.
2: For motivated beginners, harder questions from the previous categories.
2.5: More advanced beginner problems, hardest questions from previous categories.
3: Early intermediate problems that require more creative thinking.
3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge.
4: Intermediate-level problems.
4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions.
5: More difficult AIME problems or simple proof-based Olympiad-style problems.
6: High-level AIME-styled questions or introductory Olympiad-level questions.
7: Tougher Olympiad-level questions, may require more technical knowledge.
8: High-level Olympiad-level questions.
9: Expert Olympiad-level questions.
9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve.
10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
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Problem
Consider all 1000-element subsets of the set . From each such subset choose the least element. The arithmetic mean of all of these least elements is , where and are relatively prime positive integers. Find . Hint Use the Hockey Stick Identity in the form $$\binom{a}{a} + \binom{a+1}{a} + \binom{a+2}{a} + \dots + \binom{b}{a} = \binom{b+1}{a+1}.$$ (This is best proven by a combinatorial argument that coincidentally pertains to the problem: count two ways the number of subsets of the first numbers with elements whose least element is , for .)