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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 .
2. Let us assume that is a positive integral number greater than two. Find the maximum value for and the minimum value for such that
holds for any positive numbers , , , .
3. and play the following game. They arbitrarily select from among the first positive integral numbers ones. If the sum of the selected numbers is even then wins, if their sum is odd then is the winner. For what values of are equal the chances for and ?