Berkeley Math Circle Monthly Contest 3 #0hxa

Berkeley Math Circle Monthly Contest 3Hard

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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 $k$ be a positive integer, and let $(a_{1}, a_{2}, \ldots, a_{2k})$ and $(b_{1}, b_{2}, \ldots, b_{2k})$ be two sequences of real numbers such that $\frac{1}{2} \leq a_{1} \leq \cdots \leq a_{2k}$ and $\frac{1}{2} \leq b_{1} \leq \cdots \leq b_{2k}$ . Let $M$ and $m$ be the maximum and minimum respectively of

(a_{1}+c_{1})(a_{2}+c_{2}) \cdots (a_{2k}+c_{2k})

as $(c_{1}, \ldots, c_{2k})$ ranges through all possible permutations of $(b_{1}, \ldots, b_{2k})$ . Prove that

M-m \geq k(a_{k}-a_{k+1})(b_{k}-b_{k+1}) .