1974 IMO Problems
Problems of the 16th IMO 1974 in East Germany.
Three players and play the following game: On each of three cards an integer is written. These three numbers satisfy . The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated by the card he holds. Then the cards are shuffled again; the counters remain with the players.
This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round, has 20 counters in all, has 10 and has 9. At the last round received counters. Who received counters on the first round?
In the triangle , prove that there is a point on side such that is the geometric mean of and if and only if
Prove that the number is not divisible by 5 for any integer .
Consider decompositions of an chessboard into p non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If is the number of white squares in the -th rectangle, then .
Find the maximum value of for which such a decomposition is possible. For this value of , determine all possible sequences .
Determine all possible values of where are arbitrary positive numbers.
Let be a non-constant polynomial with integer coefficients. If is the number of distinct integers such that , prove that , where denotes the degree of the polynomial .
- 1974 IMO
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