2016 AMC 12B Problems/Problem 11
Problem
The grid below contains the 
points whose 
- and 
-coordinates are in the set 
:A square with all four of its vertices among these points has area 
. What is the sum of all possible values of ?
Solution
Solution by e_power_pi_times_i Revised by Kinglogic
![[asy] Label l; l.p=fontsize(8); xaxis(-1,8,Ticks(l, 1.0)); yaxis(-1,16,Ticks(l, 1.0)); real f(real x) { return x * pi; } D(graph(f,-1/pi,5.1)); D((5.1,-1)--(5.1,16)); D((-1,-0.1)--(8,-0.1)); for(int x = 0; x < 5.1; ++x) { for(int y = 0; y < 16; ++y) { if(x * pi > y) { D((x,y)); } } } [/asy]](http://latex.artofproblemsolving.com/a/b/c/abc2430047d924cb59d8f9799986b35cea9a979a.png)
(red shows lattice points within the triangle)
If we draw a picture showing the triangle, we see that it would be easier to count the squares vertically and not horizontally. The upper bound is squares 
, and the limit for the -value is 
squares. First we count the 
squares. In the back row, there are 
squares with length 
because 
generates squares from 
to 
, and continuing on we have 
, 
, and 
for -values for , 
, and in the equation 
. So there are 
squares with length in the figure. For 
squares, each square takes up units left and units up. Squares can also overlap. For squares, the back row stretches from 
to 
, so there are 
squares with length in a by box. Repeating the process, the next row stretches from 
to 
, so there are squares. Continuing and adding up in the end, there are 
squares with length in the figure. Squares with length in the back row start at and end at , so there are 
such squares in the back row. As the front row starts at 
and ends at 
there are 
squares with length . As squares with length would not fit in the triangle, the answer is 
which is 
.
See Also
| 2016 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 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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