Berkeley Math Circle Take-Home Contest #6 #0hn8

Berkeley Math Circle Take-Home Contest #6Hard

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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 $d$ be a fixed positive integer. Prove that there exists a unique polynomial $S(n)$ such that for every integer $n \geq 0$ , $S(n) = \sum_{k=0}^{n} k^{d} = 0^{d} + 1^{d} + \cdots + n^{d}$ . Also prove that $S(n)$ can be expressed in the form

c_{0} + c_{1}(2n+1) + c_{2}(2n+2)^{2} + \cdots + c_{d+1}(2n+1)^{d+1}

where the $c_{i}$ are rational numbers such that $c_{i} c_{i+1} = 0$ for $i = 0, 1, \ldots, d$ .