MathDB

Problem 5

Part of 2024 Ukraine National Mathematical Olympiad

Problems(3)

Best Ukrainian Author strikes with another genius algebra...

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 8.5

3/20/2024
Real numbers a,b,ca, b, c are such that
a2+c−bc=b2+a−ca=c2+b−aba^2+c-bc = b^2+a-ca = c^2+b-ab
Does it follow that a=b=ca=b=c?
Proposed by Mykhailo Shtandenko
algebra
Maestro composed an algebra so good we used it in two grades!

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 9.5, 10.5

3/20/2024
For real numbers a,b,c,d∈[0,1]a, b, c, d \in [0, 1], find the largest possible value of the following expression:
a2+b2+c2+d2−ab−bc−cd−daa^2+b^2+c^2+d^2-ab-bc-cd-da
Proposed by Mykhailo Shtandenko
algebraInequalityinequalities proposedinequalities
A bit of casework but imo pleasant

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.5

3/20/2024
You are given some 1212 non-zero, not necessarily distinct real numbers. Find all positive integers kk from 11 to 1212, such that among these numbers you can always choose kk numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative.
Proposed by Anton Trygub
algebra