Difference between revisions of "1995 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
| − | Given that <math> | + | Given that <math>(1+\sin t)(1+\cos t)=5/4</math> and |
:<math>(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},</math> | :<math>(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},</math> | ||
| − | where <math> | + | where <math>k, m,</math> and <math>n_{}</math> are [[positive integer]]s with <math>m_{}</math> and <math>n_{}</math> [[relatively prime]], find <math>k+m+n.</math> |
== Solution == | == Solution == | ||
Revision as of 16:50, 28 July 2011
Problem
Given that 
and
where 
and 
are positive integers with 
and relatively prime, find 
Solution
From the givens,
, and adding 
to both sides gives 
. Completing the square on the left in the variable 
gives 
. Since 
, we have 
. Subtracting twice this from our original equation gives 
, so the answer is 
.
See also
| 1995 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
