Problem:
Find the number of ordered pairs (x,y), where both x and y are integers between −100 and 100, inclusive, such that 12x2−xy−6y2=0.
Solution:
We start with the equation
12x2−xy−6y2=0.
Solving for x (or y) yields
x=24y±17∣y∣.
This splits into two families of solutions:
- x=43y: For x to be an integer, y must be divisible by 4. Write y=4k; then x=3k. Since ∣4k∣≤100 requires ∣k∣≤25, there are 51 choices for k.
- x=−32y: For x to be an integer, y must be divisible by 3. Write y=3k; then x=−2k. Since ∣3k∣≤100 requires ∣k∣≤33, there are 67 choices for k.
The solution (0,0) appears in both families, so we subtract one duplicate.
Thus, the total number of solutions is:
51+67−1=117.
Therefore, the answer is 117.
The problems on this page are the property of the MAA's American Mathematics Competitions