Difference between revisions of "1990 AIME Problems/Problem 3"
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+ | == Solution 3 == | ||
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+ | As in above, we have | ||
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+ | == Solution 3 == | ||
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+ | As in above, we have <math>rs = 118r - 116s.</math> This means that <math>rs + 116s - 118r = 0.</math> Using SFFT we obtain <math>s(r+116) - 118(r+116) = -118 \cdot 116 \implies </math>(s-118)(r+116) = -118 \cdot 116.<math> Since </math>r+116<math> is always positive, we know thta </math>s-118<math> must be negative. Therefore the maximum value of </math>s<math> must be </math>\boxed{117}<math> which indeed yields an integral value of </math>r.$ | ||
== See also == | == See also == |
Revision as of 20:52, 20 September 2020
Problem
Let be a regular and be a regular such that each interior angle of is as large as each interior angle of . What's the largest possible value of ?
Solution 1
The formula for the interior angle of a regular sided polygon is .
Thus, . Cross multiplying and simplifying, we get . Cross multiply and combine like terms again to yield . Solving for , we get .
and , making the numerator of the fraction positive. To make the denominator positive, ; the largest possible value of is .
This is achievable because the denominator is , making a positive number and .
Solution 2
Like above, use the formula for the interior angles of a regular sided polygon.
This equation tells us divides . If specifically divides 118 then the highest it can be is 118. However, this gives an equation with no solution. The second largest possibility in this case is , which does give a solution: . Although, the problem asks for , not . The only conceivable reasoning behind this is that is greater than 1000. This prompts us to look into the second case, where divides . Make . Rewrite the equation using this new information.
Now we now k divides 116. The larger k is, the larger s will be, so we set k to be the maximum: 116.
-jackshi2006
Solution 3
As in above, we have
Solution 3
As in above, we have This means that Using SFFT we obtain (s-118)(r+116) = -118 \cdot 116.r+116s-118s\boxed{117}r.$
See also
1990 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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