MathDB

2

Part of 2023 Kazakhstan National Olympiad

Problems(2)

Inequality x^2+y^2+z^2=2xyz+1

Source: Kazakhstan Junior Olympiad 2023 , 9 grade / Problem 2

3/21/2023
a,b,ca,b,c are positive real numbers such that a+b+c≥3a+b+c\ge 3 and a2+b2+c2=2abc+1a^2+b^2+c^2=2abc+1. Prove that a+b+c≤2abc+1a+b+c\le 2\sqrt{abc}+1
inequalities
Weird algebra with combinatorial flavour

Source: Kazakhstan National MO 2023 (10-11).2

3/21/2023
Let n>100n>100 be an integer. The numbers 1,2…,4n1,2 \ldots, 4n are split into nn groups of 44. Prove that there are at least (n−6)22\frac{(n-6)^2}{2} quadruples (a,b,c,d)(a, b, c, d) such that they are all in different groups, a<b<c<da<b<c<d and c−b≤∣ad−bc∣≤d−ac-b \leq |ad-bc|\leq d-a.
algebra