Berkeley Math Circle Take-Home Contest #0hjr

Berkeley Math Circle Take-Home ContestHard

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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.

Linear and quadratic inequalitiesVectorsInduction / smoothing

Problem

Problem:

Given are $n+1$ real linear equations in $n$ variables (of the form $a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}=a$ ). Prove that each $=$ sign can be replaced with either $\leq$ or $\geq$ so that the resulting $n+1$ inequalities have the following property: for every choice of real numbers $x_{1}, x_{2}, \ldots, x_{n}$ , at least one inequality is true.

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