Think and solve!

Since 1974, when the traditional Delta made its first public appearance, over 840 mathematical problems and over 470 problems in physics have occupied our readers' minds, not counting hundreds of problems included in our continuous mathematical and physical leagues. In most cases their solution requires more thinking than computing, and finding an elegant solution can provide much satisfaction.

The problems are always published with a solution, hidden somewhere in the same issue of the journal. However, we will not provide solutions here. Try for yourself. And if you really cannot find anything and you are sure you do not want to continue trying, but you still wish to know the solution, write to or and ask for one. In order to make our communication easier, we preserve the original numbers of the problems as they appeared in Delta. You may quote them when asking for a solution. But before you ask, think and solve!

In this issue we offer you three mathematical problems by Krzysztof Oleszkiewicz (M 824, M 825) and Michal Wojciechowski (M 823).

M 823. Non-zero vectors v1, v2, …, vn are given in the three-dimensional space and the angle between any two of them is obtuse. Prove that n is not bigger than 4.

M 824. Assume that some three orthogonal sections of a convex polyhedron which has a center of symmetry are squares. Does that necessarily imply that the polyhedron is a cube?

M 825. Prove that if p>1, and x1, x2, …, xn are positive numbers, then