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A1. Three players play the following game. There are three cards each with a different positive integer. In each round the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received 20, another 10 and the third 9 counters. In the last round the player with 10 received the largest number of counters. Who received the middle number on the first round?
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A2. Prove that there is a point D on the side AB of the triangle ABC, such that CD is the geometric mean of AD and DB if and only if sin A sin B ≤ sin2(C/2).
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A3. Prove that the sum from k = 0 to n of (2n+1)C(2k+1) 23k is not divisible by 5 for any non-negative integer n. [rCs denotes the binomial coefficient r!/(s!(r-s)!) .]
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B1. An 8 x 8 chessboard is divided into p disjoint rectangles (along the lines between squares), so that each rectangle has the same number of white squares as black squares, and each rectangle has a different number of squares. Find the maximum possible value of p and all possible sets of rectangle sizes.
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B2. Determine all possible values of a/(a+b+d) + b/(a+b+c) + c/(b+c+d) + d/(a+c+d) for positive reals a, b, c, d.
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B3. Let P(x) be a polynomial with integer coefficients of degree d > 0. Let n be the number of distinct integer roots to P(x) = 1 or -1. Prove that n ≤ d + 2.
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