Difference between revisions of "1985 AIME Problems/Problem 4"
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[[Image:AIME_1985_Problem_4.png]] | [[Image:AIME_1985_Problem_4.png]] | ||
| − | == Solution == | + | == Solution 1== |
| − | 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 = | + | 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 = \boxed{032}</math>. |
==Solution 2== | ==Solution 2== | ||
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n(n-1)=992</math> | n(n-1)=992</math> | ||
<div style="text-align:center;"> | <div style="text-align:center;"> | ||
| − | Simple factorization and guess and check gives us <math>\boxed{ | + | Simple factorization and guess and check gives us <math>\boxed{032}</math>. |
== See also == | == See also == | ||
Revision as of 22:45, 20 September 2011
Contents
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 1
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)
.
