MathDB

2

Part of 1998 National High School Mathematics League

Problems(2)

Set Problem

Source: 1998 National High School Mathematics League, Exam One, Problem 2

3/8/2020
Nonempty set A={x∣2a+1≤x≤3a−5},B={x∣3≤x≤22}A=\{x|2a+1\leq x\leq 3a-5\},B=\{x|3\leq x\leq22\}, then the range value of aa such that A⊆A∩BA\subseteq A\cap B is (A){a∣1≤a≤9}(B){a∣6≤a≤9}(C){a∣a≤9}(D)∅\text{(A)}\{a|1\leq a \leq9\}\qquad\text{(B)}\{a|6\leq a \leq9\}\qquad\text{(C)}\{a|a \leq9\}\qquad\text{(D)}\varnothing
Inequality

Source: 1998 National High School Mathematics League, Exam Two, Problem 2

3/9/2020
Let a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n are real numbers in [1,2][1,2]. If ∑i=1nai2=∑i=1nbi2\sum_{i=1}^{n}a_i^2=\sum_{i=1}^{n}b_i^2, prove that ∑i=1nai3bi≤1710∑i=1nai2.\sum_{i=1}^{n}\frac{a_i^3}{b_i}\leq\frac{17}{10}\sum_{i=1}^{n}a_i^2. Find when the equality holds.
inequalities