MathDB

6

Part of 2007 AIME Problems

Problems(2)

Frog on a Number Line

Source: AIME I 2007 #6

3/15/2007
A frog is placed at the origin on a number line, and moves according to the following rule: in a given move, the frog advanced to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, 0, 3, 6, 13, 15, 26, 39 is a move sequence. How many move sequences are possible for the frog?
analytic geometryAMC
Parity-Monotonic Numbers

Source: AIME II 2007 #6

3/29/2007
An integer is called parity-monotonic if its decimal representation a1a2a3⋯aka_{1}a_{2}a_{3}\cdots a_{k} satisfies ai<ai+1a_{i}<a_{i+1} if aia_{i} is odd, and ai>ai+1a_{i}>a_{i+1} is aia_{i} is even. How many four-digit parity-monotonic integers are there?
AMCAIMEAoPSwikiAIME IIAIME I