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inequality

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 3

November 15, 2009
inequalitiesfloor functionDiophantine equationnumber theory proposednumber theory

Problem Statement

For each real number x x< let ⌊x⌋ \lfloor x \rfloor be the integer satisfying \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1 and let \{x\}\equal{}x\minus{}\lfloor x \rfloor. Let c c be a real number such that {n3}>cn3 \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}} for all positive integers n n. Prove that c≤1 c \le 1.
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