MathDB

4

Part of 2012 South East Mathematical Olympiad

Problems(2)

find maximal value of a+b-c+d

Source: China south east mathematical olympiad 2012 day1 problem 4

7/17/2013
Let a,b,c,da, b, c, d be real numbers satisfying inequality acos⁡x+bcos⁡2x+ccos⁡3x+dcos⁡4x≤1a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1 holds for arbitrary real number xx. Find the maximal value of a+b−c+da+b-c+d and determine the values of a,b,c,da,b,c,d when that maximum is attained.
inequalitiestrigonometryinequalities unsolved
problem of grasshopper leaping

Source: China south east mathematical olympiad 2012 day2 problem 8

7/17/2013
Let positive integers m,nm,n satisfy n=2m−1n=2^m-1. Pn={1,2,⋯ ,n}P_n =\{1,2,\cdots ,n\} is a set that contains nn points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of mm satisfying following conditions: (a) x,yx, y are two arbitrary points in PnP_n; (b) starting at point xx, the grasshopper leaps 20122012 times and finishes at point yy; (the grasshopper is allowed to travel xx and yy more than once) (c) there are even number ways for the grasshopper to do (b).
combinatorics unsolvedcombinatorics