MathDB

18

Part of 2014 BMT Spring

Problems(2)

2014 BMT Team 18

Source:

1/6/2022
Monty wants to play a game with you. He shows you five boxes, one of which contains a prize and four of which contain nothing. He allows you to choose one box but not to open it. He then opens one of the other four boxes that he knows to contain nothing. Then, he makes you switch and choose a different, unopened box. However, Monty sketchily reveals the contents of another (empty) box, selected uniformly at random from the two or three closed boxes (that you do not currently have chosen) that he knows to contain no prize. He then offers you the chance to switch again. Assuming you seek to maximize your return, determine the probability you get a prize.
combinatorics
BMT 2014 Spring - Individual 18

Source:

1/22/2022
Suppose the polynomial f(x)=x2014f(x) = x^{2014} is equal to f(x)=∑k=02014ak(xk)f(x) =\sum^{2014}_{k=0} a_k {x \choose k} for some real numbers a0,...,a2014a_0,... , a_{2014}. Find the largest integer mm such that 2m2^m divides a2013a_{2013}.
number theory