MathDB

3

Part of 1998 China Team Selection Test

Problems(2)

Inequality involving sines and cosines

Source: China TST 1998, problem 3

5/22/2005
For a fixed θ∈[0,π2]\theta \in \lbrack 0, \frac{\pi}{2} \rbrack, find the smallest a∈R+a \in \mathbb{R}^{+} which satisfies the following conditions: I. acos⁡θ+asin⁡θ>1\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1. II. There exists x∈[1−asin⁡θ,acos⁡θ]x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack such that [(1−x)sin⁡θ−a−x2cos⁡2θ]2+[xcos⁡θ−a−(1−x)2sin⁡2θ]2≤a\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a.
inequalitiestrigonometryinequalities unsolved
China TST 1998 division probelm

Source: China TST 1998, problem 6

5/22/2005
For any h=2rh = 2^{r} (rr is a non-negative integer), find all k∈Nk \in \mathbb{N} which satisfy the following condition: There exists an odd natural number m>1m > 1 and n∈Nn \in \mathbb{N}, such that k∣mh−1,m∣nmh−1k+1k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1.
inductionLaTeXnumber theory unsolvednumber theory