Difference between revisions of "1997 AIME Problems/Problem 2"
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Thus <math>\frac sr = \dfrac{204}{1296}=\dfrac{17}{108}</math>, and <math>m+n=\boxed{125}</math>. | Thus <math>\frac sr = \dfrac{204}{1296}=\dfrac{17}{108}</math>, and <math>m+n=\boxed{125}</math>. | ||
| + | |||
| + | == Solution == | ||
| + | First, to find the number of squares, we can look case by case by the side length of the possible squares on the checkerboard. We see that there are <math>8^2</math> ways to place a <math>1</math> x <math>1</math> square and <math>7^2</math> for a <math>2</math> x <math>2</math> square. This pattern can be easily generalized and we see that the number of squares is just <math>\sum^8_{i=1}{i^2}</math>. This can be simplified by using the well-known formula for the sum of consecutive squares <math>\frac{n(n+1)(2n+1)}{6}</math> to get <math>204</math>. | ||
| + | |||
| + | Then, to find the number of rectangles, first note that a square falls under the definition of a rectangle. We can break up the rectangles into cases for the length x width. As we note down the cases for <math>1</math>x<math>1, 1</math>x<math>2 , 2</math>x<math>1, 2</math>x<math>2,...,</math> we see they are respectively <math>8</math>x<math>8, 8</math>x<math>7, 7</math>x<math>8, 7</math>x<math>7, ...</math>. We can quickly generalize this pattern to basically just <math>{\sum^8_{i=1}{i}}\cdot{\sum^8_{i=1}{i}}</math>. This gets us <math>{(\frac{9\cdot8}{2})}^2,</math> which is just <math>1296.</math> | ||
| + | |||
| + | Now, to calculate the ratio of <math>s/r,</math> we divide <math>204</math> by <math>1296</math> to get a simplified fraction of <math>\frac{17}{108}.</math> | ||
| + | |||
| + | Thus, our answer is just <math>s + r = 17+108 = \boxed{125}</math> | ||
| + | ~MathWhiz35 | ||
== See also == | == See also == | ||
Revision as of 10:30, 18 February 2021
Contents
Problem
The nine horizontal and nine vertical lines on an 
checkerboard form 
rectangles, of which 
are squares. The number 
can be written in the form 
where 
and 
are relatively prime positive integers. Find 
Solution
To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or 
. Similarily, there are 
ways to pick the vertical sides, giving us 
rectangles.
For , there are 
unit squares, 
of the 
squares, and so on until 
of the 
squares. Using the sum of squares formula, that gives us 
.
Thus 
, and 
.
Solution
First, to find the number of squares, we can look case by case by the side length of the possible squares on the checkerboard. We see that there are ways to place a 
x square and for a 
x square. This pattern can be easily generalized and we see that the number of squares is just 
. This can be simplified by using the well-known formula for the sum of consecutive squares 
to get 
.
Then, to find the number of rectangles, first note that a square falls under the definition of a rectangle. We can break up the rectangles into cases for the length x width. As we note down the cases for x
x
x
x
we see they are respectively 
x
x
x
x
. We can quickly generalize this pattern to basically just 
. This gets us 
which is just 
Now, to calculate the ratio of 
we divide by 
to get a simplified fraction of 
Thus, our answer is just 
~MathWhiz35
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
