MathDB

2

Part of 2017 AIME Problems

Problems(2)

American Remainder Theorem

Source: 2017 AIME I #2

3/8/2017
When each of 702, 787, and 855 is divided by the positive integer mm, the remainder is always the positive integer rr. When each of 412, 722, and 815 is divided by the positive integer nn, the remainder is always the positive integer s≠rs \neq r. Fine m+n+r+sm+n+r+s.
AMCAIME2017 AIME I
Pride and Prejudisemifinals

Source: 2017 AIME II #2

3/23/2017
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.
AMCAIMEAIME II