MathDB

3

Part of 2003 China National Olympiad

Problems(2)

If product of tan x_i is 2^(n/2) then inequality holds

Source: Chinese Mathematical Olympiad 2003 Problem 3

2/18/2012
Given a positive integer nn, find the least λ>0\lambda>0 such that for any x1,…xn∈(0,π2)x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right), the condition ∏i=1ntan⁡xi=2n2\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}} implies ∑i=1ncos⁡xi≤λ\sum_{i=1}^{n}\cos x_i\le\lambda.
Huang Yumin
inequalitiestrigonometryinequalities proposed
Inequality with reals a,b,c,d and x_i, y_i

Source: Chinese Mathematical Olympiad 2003 Problem 6

2/18/2012
Suppose a,b,c,da,b,c,d are positive reals such that ab+cd=1ab+cd=1 and xi,yix_i,y_i are real numbers such that xi2+yi2=1x_i^2+y_i^2=1 for i=1,2,3,4i=1,2,3,4. Prove that (ax1+bx2+cx3+dx4)2+(ay4+by3+cy2+dy1)2≤2(a2+b2ab+c2+d2cd).(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).
Li Shenghong
inequalitiesinequalities proposed