Difference between revisions of "1986 AIME Problems/Problem 4"
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== Solution == | == Solution == | ||
Adding all five [[equation]]s gives us <math>6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)</math> so <math>x_1 + x_2 + x_3 + x_4 + x_5 = 31</math>. Subtracting this from the fourth given equation gives <math>x_4 = 17</math> and subtracting it from the fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\cdot17 + 2\cdot65 = \boxed{181}</math>. | Adding all five [[equation]]s gives us <math>6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)</math> so <math>x_1 + x_2 + x_3 + x_4 + x_5 = 31</math>. Subtracting this from the fourth given equation gives <math>x_4 = 17</math> and subtracting it from the fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\cdot17 + 2\cdot65 = \boxed{181}</math>. | ||
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| + | == Solution 2 == | ||
== See also == | == See also == | ||
Revision as of 15:05, 22 August 2019
Contents
Problem
Determine 
if 
, 
, 
, 
, and 
satisfy the system of equations below.
Solution
Adding all five equations gives us 
so 
. Subtracting this from the fourth given equation gives 
and subtracting it from the fifth given equation gives 
, so our answer is 
.
Solution 2
See also
| 1986 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
