Berkeley Math Circle Monthly Contest 4 #0hru

Berkeley Math Circle Monthly Contest 4Hard

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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 $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers whose sum is equal to $1$ . If

\begin{aligned} S= & \frac{a_{1}^{2}}{2 a_{1}}+\frac{a_{1} a_{2}}{a_{1}+a_{2}}+\frac{a_{1} a_{3}}{a_{1}+a_{3}}+\cdots+\frac{a_{1} a_{n}}{a_{1}+a_{n}} \\ & +\frac{a_{2} a_{1}}{a_{2}+a_{1}}+\frac{a_{2}^{2}}{2 a_{2}}+\frac{a_{2} a_{3}}{a_{2}+a_{3}}+\cdots+\frac{a_{2} a_{n}}{a_{2}+a_{n}} \\ & \vdots \\ & +\frac{a_{n} a_{1}}{a_{n}+a_{1}}+\frac{a_{n} a_{2}}{a_{n}+a_{2}}+\frac{a_{n} a_{3}}{a_{n}+a_{3}}+\cdots+\frac{a_{n}^{2}}{2 a_{n}} \end{aligned}

prove that $S \leq \frac{n}{2}$ .