MathDB

2

Part of 2022 AIME Problems

Problems(2)

I am based

Source: AIME I #2

2/9/2022
Find the three-digit positive integer a‾ b‾ c‾\underline{a} \ \underline{b} \ \underline{c} whose representation in base nine is b‾ c‾ a‾nine\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}, where aa, bb, and cc are (not necessarily distinct) digits.
AMCAIMEAIME I
Another year another tournament

Source: 2022 AIME II Problem 2

2/17/2022
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability 23\frac23. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability 34\frac34. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
AIMEAIME II