Prove that there is no graph with five vertices whose degrees are equal to 4, 4, 4, 4, 2.
Prove that a graph, in which every two vertices are connected by exactly one simple path, is a tree.
Prove that, in a tree, every two vertices are connected by exactly one simple path.
Eugenie, arriving from Big-island, said that there are several lakes connected by rivers. Three rivers flow from each lake, and four rivers flow into each lake. Prove that she is wrong.
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).
A class has more than 32, but less than 40 people. Every boy is friends with three girls, and every girl is friends with five boys. How many people are there in the class?
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).