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Found: 3

Problem #PRU-57309

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality algebra problems

13–13 2.0

$a$ , $b$ 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.

Problem #PRU-57310

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality algebra problems

13–13 2.0

a, b and c are the lengths of the sides of an arbitrary triangle. Prove that $a^{2} + b^{2} + c^{2} < 2 (a b + b c + c a)$ .

Problem #PRU-110207

Problems Methods Pigeonhole principle Pigeonhole principle (other) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality algebra problems

12–14 3.0

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.