Difference between revisions of "1985 AIME Problems/Problem 4"
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== Problem == | == Problem == | ||
| − | A small [[square (geometry) | square]] is constructed inside a square of [[area]] 1 by dividing each side of the unit square into <math>n</math> equal parts, and then connecting the [[vertex | vertices]] to the division points closest to the opposite vertices. Find the value of <math>n</math> if the the area of the small square is exactly <math>\frac1{1985}</math>. | + | A small [[square (geometry) | square]] is constructed inside a square of [[area]] 1 by dividing each side of the unit square into <math>n</math> equal parts, and then connecting the [[vertex | vertices]] to the division points closest to the opposite vertices. Find the value of <math>n</math> if the the [[area]] of the small square is exactly <math>\frac1{1985}</math>. |
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== Solution == | == Solution == | ||
| − | {{ | + | The lines passing through <math>A</math> and <math>C</math> divide the square into three parts, two [[right triangle]]s and a [[parallelogram]]. The area of the [[triangle]]s together is easily seen to be <math>\frac{n - 1}{n}</math>, so the area of the parallelogram is <math>A = \frac{1}{n}</math>. By the [[Pythagorean Theorem]], the base of the parallelogram has [[length]] <math>l = \sqrt{1^2 + \left(\frac{n - 1}{n}\right)^2} = \frac{1}{n}\sqrt{2n^2 - 2n + 1}</math>, so the parallelogram has height <math>h = \frac{A}{l} = \frac{1}{\sqrt{2n^2 - 2n + 1}}</math>. But the height of the parallelogram is the side of the little square, so <math>2n^2 - 2n + 1 = 1985</math>. Solving this [[quadratic equation]] gives <math>n = 032</math>. |
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== See also == | == See also == | ||
* [[1985 AIME Problems/Problem 3 | Previous problem]] | * [[1985 AIME Problems/Problem 3 | Previous problem]] | ||
Revision as of 18:09, 19 January 2007
Problem
A small square is constructed inside a square of area 1 by dividing each side of the unit square into 
equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of if the the area of the small square is exactly 
.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Solution
The lines passing through 
and 
divide the square into three parts, two right triangles and a parallelogram. The area of the triangles together is easily seen to be 
, so the area of the parallelogram is 
. By the Pythagorean Theorem, the base of the parallelogram has length 
, so the parallelogram has height 
. But the height of the parallelogram is the side of the little square, so 
. Solving this quadratic equation gives 
.
