MathDB

5

Part of 2021 Caucasus Mathematical Olympiad

Problems(2)

The product of pairwise gcd-s is a perfect square

Source: VI Caucasus Mathematical Olympiad

3/14/2021
Let a,b,ca, b, c be positive integers such that the product gcd⁡(a,b)⋅gcd⁡(b,c)⋅gcd⁡(c,a)\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) is a perfect square. Prove that the product lcm⁡(a,b)⋅lcm⁡(b,c)⋅lcm⁡(c,a)\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) is also a perfect square.
number theory
No triangle with heights as sides

Source: VI Caucasus Mathematical Olympiad

3/14/2021
A triangle Δ\Delta with sidelengths a≤b≤ca\leq b\leq c is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of Δ\Delta. Prove that b2>acb^2>ac.
geometryGeometry inequality