MathDB

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Part of 2006 All-Russian Olympiad

Problems(2)

Symmetric non-self-intersecting path in 15*15 chessboard

Source: All-Russian Olympiad 2006 finals, problem 10.1 = 9.1

5/7/2006
Given a 15×1515\times 15 chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is ≤200\leq 200.
combinatorics proposedcombinatorics
Sin sqrt(x) < sqrt(sin x) for 0 < x < pi/2

Source: All-Russian Olympiad 2006 finals, problem 11.1

5/6/2006
Prove that sin⁡x<sin⁡x\sin\sqrt{x}<\sqrt{\sin x} for every real xx such that 0<x<π20<x<\frac{\pi}{2}.
trigonometryfunctioncalculusderivativeinequalitiesreal analysisreal analysis unsolved