MathDB

1

Part of 2017 Serbia National Math Olympiad

Problems(2)

Inequality with a+b+c = 1 [Serbia MO 2017, D1, P1]

Source: Serbia National Mathematical Olympiad 2017, Day 1, Problem 1

4/1/2017
Prove that for positive real numbers a,b,ca,b,c such that a+b+c=1a+b+c=1, a2b+1+b2c+1+c2a+1≤2−(a2+b2+c2).a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.
inequalities
Specific divisors implying perfect square

Source: Serbian MO 2017 4

4/2/2017
Let aa be a positive integer.Suppose that ∀n\forall n ,∃d\exists d, d≠1d\not =1, d≡1(modn)d\equiv 1\pmod n ,d∣n2a−1d\mid n^2a-1.Prove that aa is a perfect square.
modular arithmetic