MathDB

2

Part of 2000 National High School Mathematics League

Problems(2)

Trigonometry

Source: 2000 National High School Mathematics League, Exam One, Problem 2

3/10/2020
If sin⁡α>0,cos⁡α<0,sin⁡α3>cos⁡α3\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}, then the range value of α3\frac{\alpha}{3} is (A)(2kπ+π6,2kπ+π3),k∈Z\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z} (B)(2kπ3+π6,2kπ3+π3),k∈Z\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z} (C)(2kπ+5π6,2kπ+π),k∈Z\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z} (D)(2kπ+π4,2kπ+π3)∪(2kπ+5π6,2kπ+π),k∈Z\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}
trigonometry
Number Theory

Source: 2000 National High School Mathematics League, Exam Two, Problem 2

3/11/2020
Two sequences (an)(a_n) and (bn)(b_n) satisfy that a0=1,a1=4,a2=49a_0=1,a_1=4,a_2=49, and {an+1=7an+6bn−3bn+1=8an+7bn−4\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases} for n=0,1,2,⋯ ,n=0,1,2,\cdots,. Prove that ana_n is a perfect square for n=0,1,2,⋯ ,n=0,1,2,\cdots,.
number theory