MathDB

2

Part of 1983 Vietnam National Olympiad

Problems(2)

Two trigonometric inequalities.

Source: Vietnam MO 1983 P2

3/18/2011
(a)(a) Prove that 2(sin⁡t+cos⁡t)≥2sin⁡2t4\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t} for 0≤t≤π2.0 \le t \le\frac{\pi}{2}. (b)(b) Find all y,0<y<πy, 0 < y < \pi, such that 1+2cot⁡2ycot⁡y≥tan⁡2ytan⁡y1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}. .
trigonometryinequalities unsolvedinequalities
Finding the larger of two algebraic expressions

Source: Vietnam MO 1983 P5

3/18/2011
Decide whether SnS_n or TnT_n is larger, where Sn=∑k=1nk(2n−2k+1)(2n−k+1),Tn=∑k=1n1kS_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}
inequalities proposedinequalities