Difference between revisions of "2020 AMC 12A Problems/Problem 17"
Lopkiloinm (talk | contribs) (→Solution 1) |
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Let the <math>x</math>-intercept be <math>n.</math> | Let the <math>x</math>-intercept be <math>n.</math> | ||
| − | Realize that by the shoelace formula the area of the triangle must be <math>-\ln{n}+\ln{n+1}+\ln{n+2}-\ln{n+3}.</math> That equals to <math>\ln\frac{(n+1)(n+2)}{ | + | Realize that by the shoelace formula the area of the triangle must be <math>-\ln{n}+\ln{n+1}+\ln{n+2}-\ln{n+3}.</math> That equals to <math>\ln\frac{(n+1)(n+2)}{n(n+3)}.</math> |
<math>\ln\frac{(n+1)(n+2)}{n(n+3)} = \ln\frac{n^{2}+3n+2}{n^{2}+3n}</math> | <math>\ln\frac{(n+1)(n+2)}{n(n+3)} = \ln\frac{n^{2}+3n+2}{n^{2}+3n}</math> | ||
Revision as of 01:02, 2 February 2020
Problem 17
The vertices of a quadrilateral lie on the graph of 
, and the 
-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is 
. What is the -coordinate of the leftmost vertex?
Solution 1
Let the -intercept be 
Realize that by the shoelace formula the area of the triangle must be 
That equals to 
The -intercept is 
~lopkiloinm.
