MathDB

2

Part of 1996 National High School Mathematics League

Problems(2)

Geometrical Sequence

Source: 1996 National High School Mathematics League, Exam One, Problem 2

3/3/2020
For geometrical sequence (an)(a_n), the first term a1=1536a_1=1536, common ratio q=−12q=-\frac{1}{2}. Let πn=∏i=1nai\pi_n=\prod_{i=1}^n a_i, so the lagerest one in (πn)(\pi_n) is (A)π9(B)π11(C)π12(D)π13\text{(A)} \pi_9\qquad\text{(B)} \pi_{11}\qquad\text{(C)} \pi_{12}\qquad\text{(D)} \pi_{13}
ratio
Trigonometry

Source: 1996 National High School Mathematics League, Exam Two, Problem 2

3/4/2020
Find the range value of aa, satisfyin that ∀x∈R,θ∈[0,π2]\forall x\in\mathbb{R},\theta\in\left[0,\frac{\pi}{2}\right], (x+3+2sin⁡θcos⁡θ)2+(x+asin⁡θ+acos⁡θ)2≥18.(x+3+2\sin\theta\cos\theta)^2+(x+a\sin\theta+a\cos\theta)^2\geq\frac{1}{8}.
trigonometry