1994 AIME Problem 7

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

List all integer points on

x^2+y^2=50

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

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.

Problem 7 in other years