How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).
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.
In what number system is the equality \(3 \times 4 = 10\) correct?
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(a, b, c > 0\), the following inequality is valid: \(\left(\frac{a+b+c}{3}\right)^2 \ge \frac{ab+bc+ca}{3}\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).