Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.
\(a\), \(b\) and \(c\) are the lengths of the sides of an arbitrary triangle. Prove that \(a = y + z\), \(b = x + z\) and \(c = x + y\), where \(x\), \(y\) and \(z\) are positive numbers.
a, b and c are the lengths of the sides of an arbitrary triangle. Prove that \(a^2 + b^2 + c^2 < 2 (ab + bc + ca)\).