MathDB

2

Part of 1989 National High School Mathematics League

Problems(2)

Anti-trigonometric Function

Source: 1989 National High School Mathematics League, Exam One, Problem 2

2/25/2020
Range of function f(x)=arctan⁡x+12arcsin⁡xf(x)=\arctan x+\frac{1}{2}\arcsin x is (A)(−π,π)(B)[−34π,34π](C)(−34π,34π)(D)[−12π,12π]\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]
function
Inequality

Source: 1989 National High School Mathematics League, Exam Two, Problem 2

2/25/2020
xi∈R(i=1,2,⋯ ,n;n≥2)x_i\in\mathbb{R}(i=1,2,\cdots,n;n\geq2), satisfying that ∑i=1n∣xi∣=1,∑i=1nxi=0\sum_{i=1}^n|x_i|=1,\sum_{i=1}^nx_i=0. Prove that ∣∑i=1nxii∣≤12−12n|\sum_{i=1}^n\frac{x_i}{i}|\leq\frac{1}{2}-\frac{1}{2n}
inequalities