ITOT 45 Spring 2024 · THPT · O (Basic)

A sequence of real numbers$a_1,a_2,\ldots$ satisfies$a_{n+2}=(a_n+a_{n+1})/2$ for every$n\ge1$. Prove that all parabolas$y=x^2+a_nx+a_{n+1}$ have a common point.

An arbitrary rectangle is divided into right triangles as shown. In each right triangle a square is inscribed with one side on its hypotenuse. Which is greater: the area of the largest square, or the sum of the areas of the other three squares?

Whenever Tom cuts a pie or a piece into two pieces, they have equal weight. If he cuts a piece into more than two pieces, their weights are pairwise different. Is it true that for every integer$N\ge10$ he can obtain exactly$N$ pieces of equal weight? Pieces cannot be joined together.

Is the sum of the internal dihedral angles along the base edges of a triangular pyramid always less than the sum of the corresponding external dihedral angles?

There are45 children in a mathematics circle, and some pairs are friends. Every partition into15 triples contains at least one triple whose three members are pairwise friends. Prove that there is a partition into15 triples such that each triple contains a pair of friends.