2005 AMC 12A Problems/Problem 12
Problem
A line passes through 
and 
. How many other points with integer coordinates are on the line and strictly between 
and 
?
Solution
For convenience’s sake, we can transform to the origin and to 
(this does not change the problem). The line 
has the equation 
. The coordinates are integers if 
, so the values of 
are 
, with a total of 
coordinates.
Solution 2
The slope of the line is
so all points on the line have the form 
for some value of 
(the rise is 111 and the run is 11). Such a point has integer coordinates if and only if is an integer, and the point is strictly between and if and only if 
. Thus, there are 
points with the required property.
-Paixiao
Solution 3
We can re-write the equation in slope-intercept form (where y is on the left side). We know that the slope is 
. Then, we have 
which reduces to 
. Now, it remains to look for values of such that 
. Since 
, the only values that work are 
. Therefore, there are 
coordinates for which this is true.
~elpianista227
See also
| 2005 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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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