Difference between revisions of "2018 AIME II Problems/Problem 8"
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<math>(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}</math> | <math>(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}</math> | ||
+ | |||
+ | A diagram of the numbers: | ||
+ | |||
+ | 5 - 20 - 71 - 207 - <math>\boxed{556}</math> | ||
+ | |||
+ | 3 - 10 - 32 - 84 - 207 | ||
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+ | 2 - 5 - 14 - 32 - 71 | ||
+ | |||
+ | 1 - 2 - 5 - 10 - 20 | ||
+ | |||
+ | 1 - 1 - 2 - 3 - 5 | ||
==Solution 2== | ==Solution 2== |
Revision as of 20:28, 17 April 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 1
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 recursively.
,
,
,
A diagram of the numbers:
5 - 20 - 71 - 207 -
3 - 10 - 32 - 84 - 207
2 - 5 - 14 - 32 - 71
1 - 2 - 5 - 10 - 20
1 - 1 - 2 - 3 - 5
Solution 2
We'll refer to the moves ,
,
, and
as
,
,
, and
, respectively. Then the possible sequences of moves that will take the frog from
to
are all the permutations of
,
,
,
,
,
,
,
, and
. We can reduce the number of cases using symmetry.
Case 1:
There are possibilities for this case.
Case 2: or
There are possibilities for this case.
Case 3:
There are possibilities for this case.
Case 4: or
There are possibilities for this case.
Case 5: or
There are possibilities for this case.
Case 6:
There are possibilities for this case.
Adding up all these cases gives us ways.
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 | ||
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