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This problem trains for: SAT-I, AMC-8, GMAT.

Consider a circle with radius 5 units, centered at the origin of a rectangular system of coordinates. How many points on the circle have integer coordinates (are lattice points)?

For a point on the circle to have integer coordinates, it has to be a solution of the Diophantine equation:

equation

where R=5 is the radius of the circle.

This equation has the trivial solutions in which x = 0 or y = 0:

equation

The non-trivial solutions of this equation - if there are any - are Pythagorean triples.

For the given length of the hypotenuse (in our case, the radius of the circle is the hypotenuse and the coordinates are the legs), there is only one possible solution: (3,4,5). However, since the lengths occur squared in the equation, there is a number of points in the plane that will satisfy. Also, all the permutations of the coordinates will be solutions.

For instance, if:

equation

is a solution, then:

equation

is also a solution. This solution is distinct from the one above since the point with coordinates (4,3) is distinct from the point with coordinates (3,4).

Taking these observations into account we find the following solutions:

equation

for a total of 12 points (4 points from the trivial solution and 8 points from the non-trivial solution.