2018 AIME II Problems/Problem 8

Problem

A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$ , the frog can jump to any of the points $(x + 1, y)$ , $(x + 2, y)$ , $(x, y + 1)$ , or $(x, y + 2)$ . Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$ .

Solution

We solve this problem by working backwards. Notice, the only points the frog can be on to jump to $(4,4)$ in one move are $(2,4),(3,4),(4,2),$ and $(4,3)$ . This applies to any other point, thus we can work our way from $(0,0)$ to $(4,4)$ , recording down the number of ways to get to each point recursively. $(0,0): 1$ $(1,0)=(0,1)=1$ $(2,0)=(0, 2)=2$ $(3,0)=(0, 3)=3$ $(4,0)=(0, 4)=5$ $(1,1)=2$ , $(1,2)=(2,1)=5$ , $(1,3)=(3,1)=10$ , $(1,4)=(4,1)= 20$ $(2,2)=14, (2,3)=(3,2)=32, (2,4)=(4,2)=71$ $(3,3)=84, (3,4)=(4,3)=207$ $(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}$

2018 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2018 AIME II Problems/Problem 8

Problem

Solution