1994 AIME Problems/Problem 7
Problem
For certain ordered pairs of real numbers, the system of equations
has at least one solution, and each solution is an ordered pair of integers. How many such ordered pairs are there?
Solution
is the equation of a circle of radius , centered at the origin. The lattice points on this circle are , , and .
is the equation of a line that does not pass through the origin. (Since yields ).
So, we are looking for the number of lines which pass through either one or two of the lattice points on the circle, but do not pass through the origin.
It is clear that if a line passes through two opposite points, then it passes through the origin, and if a line passes through two non-opposite points, the it does not pass through the origin.
There are ways to pick two distinct lattice points, and thus distinct lines which pass through two lattice points on the circle. However, of these lines pass through the origin.
Since there is a unique tangent line to the circle at each of these lattice points, there are distinct lines which pass through exactly one lattice point on the circle.
Thus, there are a total of distinct lines which pass through either one or two of the lattice points on the circle, but do not pass through the origin.
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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