Harvard-MIT Mathematics Tournament #0ifx

Harvard-MIT Mathematics TournamentHard

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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.

Problem

Problem: Let $a_{1}, a_{2}, \ldots, a_{2005}$ be real numbers such that

\begin{array}{ccccccccccc} a_{1} \cdot 1 & + & a_{2} \cdot 2 & + & a_{3} \cdot 3 & + & \cdots & + & a_{2005} \cdot 2005 & = & 0 \\ a_{1} \cdot 1^{2} & + & a_{2} \cdot 2^{2} & + & a_{3} \cdot 3^{2} & + & \cdots & + & a_{2005} \cdot 2005^{2} & = & 0 \\ a_{1} \cdot 1^{3} & + & a_{2} \cdot 2^{3} & + & a_{3} \cdot 3^{3} & + & \cdots & + & a_{2005} \cdot 2005^{3} & = & 0 \\ \vdots & & \vdots & & \vdots & & & & \vdots & & \vdots \\ a_{1} \cdot 1^{2004} & + & a_{2} \cdot 2^{2004} & + & a_{3} \cdot 3^{2004} & + & \cdots & + & a_{2005} \cdot 2005^{2004} & = & 0 \end{array}

and

a_{1} \cdot 1^{2005}+a_{2} \cdot 2^{2005}+a_{3} \cdot 3^{2005}+\cdots+a_{2005} \cdot 2005^{2005}=1 .