Difference between revisions of "1985 AIME Problems/Problem 4"
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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 = 32</math>. | 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 = 32</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | [[File:Aime.png]] | ||
+ | |||
+ | Surrounding the square with area <math>\frac{1}{1985}</math> are <math>4</math> right triangles with hypotenuse <math>1</math> (sides of the large square). Thus, <math>X + \frac{1}{1985} = 1</math>, where <math>X</math> is the area of the of the 4 triangles. | ||
+ | We can thus use proportions to solve this problem. | ||
+ | <div style="text-align:center;"> | ||
+ | <math>\begin{eqnarray*} | ||
+ | \frac{GF}{BE}=\frac{CG}{CB}\implies | ||
+ | \frac{\frac{1}{\sqrt{1985}}}{BE}=\frac{\frac{1}{n}}{1}\implies | ||
+ | BE=\frac{n\sqrt{1985}}{1985}</math> | ||
+ | <div style="text-align:left;"> | ||
+ | Also, | ||
+ | <div style="text-align:center;"> | ||
+ | <math>\begin{eqnarray*} | ||
+ | \frac{BE}{1}=\frac{EC}{\frac{n-1}{n}}\implies | ||
+ | EC=\frac{\sqrt{1985}}{1985}(n-1)</math> | ||
+ | <div style="text-align:left;"> | ||
+ | Thus, | ||
+ | <div style="text-align:center;"> | ||
+ | <math>\begin{eqnarray*} | ||
+ | 2(BE)(EC)+\frac{1}{1985}=1\ | ||
+ | 2n^{2}-2n+1=1985\ | ||
+ | n(n-1)=992</math> | ||
+ | Simple factorization and guess and check gives us <math>\boxed{32}</math>. | ||
== See also == | == See also == |
Revision as of 15:33, 22 April 2011
Contents
[hide]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 .
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 .
Solution 2
Surrounding the square with area are right triangles with hypotenuse (sides of the large square). Thus, , where is the area of the of the 4 triangles. We can thus use proportions to solve this problem.
$\begin{eqnarray*} \frac{GF}{BE}=\frac{CG}{CB}\implies \frac{\frac{1}{\sqrt{1985}}}{BE}=\frac{\frac{1}{n}}{1}\implies BE=\frac{n\sqrt{1985}}{1985}$ (Error compiling LaTeX. Unknown error_msg)
Also,
$\begin{eqnarray*} \frac{BE}{1}=\frac{EC}{\frac{n-1}{n}}\implies EC=\frac{\sqrt{1985}}{1985}(n-1)$ (Error compiling LaTeX. Unknown error_msg)
Thus,
$\begin{eqnarray*} 2(BE)(EC)+\frac{1}{1985}=1\ 2n^{2}-2n+1=1985\ n(n-1)=992$ (Error compiling LaTeX. Unknown error_msg) Simple factorization and guess and check gives us .
See also
1985 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 |