MathDB

14

Part of 2016 LMT

Problems(2)

2016 LMT Individual #14

Source:

4/10/2016
Let PP and QQ be points on ACAC and ABAB, respectively, of triangle △ABC\triangle ABC such that PB=PCPB=PC and PQ⊥ABPQ\perp AB. Suppose AQQB=APPB.\frac{AQ}{QB}=\frac{AP}{PB}. Find ∠CBA\angle CBA, in degrees.
Proposed by Nathan Ramesh
2016 LMT Theme #14

Source:

4/11/2016
A ladder style tournament is held with 20162016 participants. The players begin seeded 1,2,⋯20161,2,\cdots 2016. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After 20152015 rounds, one player remains who wins the tournament. If each player has probability of 12\tfrac{1}{2} to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has pq\tfrac{p}{q} for coprime positive integers pp and qq. Find the remainder when pp is divided by 10001000.
Proposed by Nathan Ramesh