MathDB

6

Part of 2004 AIME Problems

Problems(2)

Snakelike

Source:

11/27/2005
An integer is called snakelike if its decimal representation a1a2a3⋯aka_1a_2a_3\cdots a_k satisfies ai<ai+1a_i<a_{i+1} if ii is odd and ai>ai+1a_i>a_{i+1} if ii is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
LaTeXAMCAIMEAIME II
Banana!

Source:

11/17/2005
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio 3:2:13: 2: 1, what is the least possible total for the number of bananas?
ratiomodular arithmetic