Difference between revisions of "2018 AIME II Problems/Problem 8"
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A frog is positioned at the origin of the coordinate plane. From the point <math>(x, y)</math>, the frog can jump to any of the points <math>(x + 1, y)</math>, <math>(x + 2, y)</math>, <math>(x, y + 1)</math>, or <math>(x, y + 2)</math>. Find the number of distinct sequences of jumps in which the frog begins at <math>(0, 0)</math> and ends at <math>(4, 4)</math>. | A frog is positioned at the origin of the coordinate plane. From the point <math>(x, y)</math>, the frog can jump to any of the points <math>(x + 1, y)</math>, <math>(x + 2, y)</math>, <math>(x, y + 1)</math>, or <math>(x, y + 2)</math>. Find the number of distinct sequences of jumps in which the frog begins at <math>(0, 0)</math> and ends at <math>(4, 4)</math>. | ||
| + | ==Solution== | ||
| + | We solve this problem by working backwards. Notice, the only points the frog can be on to jump to <math>(4,4)</math> in one move are <math>(2,4),(3,4),(4,2),</math> and <math>(4,3)</math>. This applies to any other point, thus we can work our way from <math>(0,0)</math> to <math>(4,4)</math>, recording down the number of ways to get to each point. | ||
| + | <math>\begin{tikzpicture} | ||
| + | \draw[step=0.5cm, color=gray] (0,0) grid(4,4); | ||
| + | \end{tikzpicture}</math> | ||
{{AIME box|year=2018|n=II|num-b=7|num-a=9}} | {{AIME box|year=2018|n=II|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 10:12, 24 March 2018
Problem
A frog is positioned at the origin of the coordinate plane. From the point 
, the frog can jump to any of the points 
, 
, 
, or 
. Find the number of distinct sequences of jumps in which the frog begins at 
and ends at 
.
Solution
We solve this problem by working backwards. Notice, the only points the frog can be on to jump to 
in one move are 
and 
. This applies to any other point, thus we can work our way from 
to , recording down the number of ways to get to each point.
$\begin{tikzpicture}
\draw[step=0.5cm, color=gray] (0,0) grid(4,4);
\end{tikzpicture}$ (Error compiling LaTeX. Unknown error_msg)
| 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. 
