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Pride and Prejudisemifinals

Source: 2017 AIME II #2

March 23, 2017
AMCAIMEAIME II

Problem Statement

Teams T1T_1, T2T_2, T3T_3, and T4T_4 are in the playoffs. In the semifinal matches, T1T_1 plays T4T_4 and T2T_2 plays T3T_3. The winners of those two matches will play each other in the final match to determine the champion. When TiT_i plays TjT_j, the probability that TiT_i wins is ii+j\frac{i}{i+j}, and the outcomes of all the matches are independent. The probability that T4T_4 will be the champion is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
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