Question

For certain ordered pairs $(a,b)$ of real numbers, the system of equations $\begin{aligned}ax+by&=1,\\x^2+y^2&=50\end{aligned}$ has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

Answer

Answer: 72

Concepts:lattice point chord tangent line

Difficulty rating: 2350

Small Hint:

List all integer points on $x^2+y^2=50$

Big Hint:

Count both chords through two non-antipodal lattice points and tangents at one lattice point

Solution:

The circle has the $12$ lattice points obtained from $(\pm1,\pm7),$ $(\pm5,\pm5),$ and $(\pm7,\pm1).$ A secant satisfying the condition is determined by any two non-antipodal lattice points. This gives $\binom{12}{2}-6=60$ lines; antipodal pairs are excluded because their line passes through the origin and cannot have equation $ax+by=1.$ There are also $12$ tangents, one at each lattice point. Each line has a unique normalization $ax+by=1,$ so the total is $60+12=72.$

1994 AIME Problem 7

Answer: 72

Difficulty rating: 2350

Small Hint:

Big Hint:

Solution:

Problem 7 in Other Years