MathDB

3

Part of 2021 Tuymaada Olympiad

Problems(2)

Numbers dividing their pairwise sums

Source: Tuymaada 2021/J3

7/30/2021
For nn distinct positive integers all their n(n−1)/2n(n-1)/2 pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?
number theory
Inequality back to business again!

Source: Tuymaada 2021 Senior P3

7/28/2021
Positive real numbers a1,…,ak,b1,…,bka_1, \dots, a_k, b_1, \dots, b_k are given. Let A=∑i=1kai,B=∑i=1kbiA = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i. Prove the inequality (∑i=1kaibiaiB+biA−1)2≥∑i=1kai2aiB+biA⋅∑i=1kbi2aiB+biA. \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}.
inequalitiesalgebra