Cím: Problems of the 1986 Kürschák József Competition
Füzet: 1987/február, 80. oldal  PDF file
Témakör(ök): Kürschák József (korábban Eötvös Loránd)

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Problems of the 1986 Kürschák József Competition
1. Prove that three semi lines starting from a given point contain three face diagonals of a cuboid if and only if the semi lines include pairwise acute angles such that their sum is 180.
2. Let us assume that n is a positive integral number greater than two. Find the maximum value for h and the minimum value for H such that

h<a1a1+a2+a2a2+a3+...+anan+a1<H,
holds for any positive numbers a1, a2, ..., an.
3. A and B play the following game. They arbitrarily select from among the first 100 positive integral numbers k ones. If the sum of the selected numbers is even then A wins, if their sum is odd then B is the winner. For what values of k are equal the chances for A and B?