1980 AMC 12 Problem 29

Question

How many ordered triples

(x,y,z)

of integers satisfy the system of equations below?

\begin{aligned} x^2-3xy+2y^2-z^2&=31,\\ -x^2+6yz+2z^2&=44,\\ x^2+xy+8z^2&=100. \end{aligned}

0

1

2

a finite number greater than two

infinitely many

Add all three equations before trying to solve for the variables

Rewrite the resulting quadratic form as a sum of two squares and reduce modulo

4

Adding the three equations gives

\begin{aligned} x^2-2xy+2y^2 &{}+6yz+9z^2\\ &=175, \end{aligned}

or

(x-y)^2+(y+3z)^2=175.

A square is congruent to

0

or

1\pmod4,

so a sum of two squares cannot be congruent to

3\pmod4.

But

175\equiv3\pmod4,

a contradiction. Thus there are no integer triples.

Therefore, the correct answer is A.

Problem 29 in other years