MathDB
Extremely Hard Inequality :>

Source: 2019 Taiwan TST Round 1

March 31, 2020
inequalities

Problem Statement

Assume a1≥a2≥⋯≥a107>0 a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 satisfy ∑k=1107ak≥M \sum\limits_{k=1}^{107}{a_{k}} \ge M and b107≥b106≥⋯≥b1>0 b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 satisfy ∑k=1107bk≤M \sum\limits_{k=1}^{107}{b_{k}} \le M . Prove that for any m∈{1,2,…,107} m \in \{1,2, \dots, 107\} , the arithmetic mean of the following numbers a1b1,a2b2,…,ambm \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}} is greater than or equal to MN \frac{M}{N}
Back to ProblemsView on AoPS