2020 AMC 12A Problems/Problem 17
Contents
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 coordinates of the quadrilateral be 
. We have by shoelace's theorem, that the area is![\[\frac{\ln(n)(n+1) + \ln(n+1)(n+2) + \ln(n+2)(n+3)+n\ln(n+3)}{2} - \frac{\ln(n+1)\ln(n) + \ln(n+2)\ln(n+1) + \ln(n+3)(n+2)+\ln(n)(n+3)}{2}=\]](http://latex.artofproblemsolving.com/d/7/c/d7c3a9dd5365eb0b3ed43de261d00ebee7be223e.png)
![\[\frac{\ln \left( \frac{n^{n+1}(n+1)^{n+2}(n+2)^{n+3}(n+3)^n}{(n+1)^n(n+2)^{n+1}(n+3)^{n+2}n^{n+3}}\right)}{2} = \ln \left( \sqrt{\frac{(n+1)^2(n+3)^2}{n^2(n+2)^2}} \right) = \ln \left(\frac{(n+1)(n+2)}{n(n+3)}\right) = \ln \left( \frac{91}{90} \right).\]](http://latex.artofproblemsolving.com/7/5/a/75afa30306f26c4e3e96ea2bbeec1cdbeedbcb8a.png)
We now that the numerator must have a factor of 
, so given the answer choices, 
is either 
or 
. If 
, the expression 
does not evaluate to 
, but if 
, the expression evaluates to . Hence, our answer is 
.
~AopsUser101
Solution 2
Like above, use the shoelace formula to find that the area of the triangle is equal to 
. Because the final area we are looking for is 
, the numerator factors into and 
, which one of 
and 
has to be a multiple of and the other has to be a multiple of . Clearly, the only choice for that is
~Solution by IronicNinja
See Also
| 2020 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 16 |
Followed by Problem 18 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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