1980 AHSME Problems/Problem 29

Problem

How many ordered triples (x,y,z) of integers satisfy the system of equations below?

$\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}$

$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \\ \text{(D)}\ \text{a finite number greater than 2}\qquad\\ \text{(E)}\ \text{infinitely many}$

Solution

Sum of three equations,

$x^2-2xy+2y^2+6yz+9z^2 = (x-y)^2+(y+3z)^2 = 175$

(x,y,z) are integers, ie. $175 = a^2 + b^2$ ,

$a^2$ : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 $b^2$ : 174, 171, 166, 159, 150, 139, 126, 111, 94, 75, 54, 31, 6

so there is NO solution

Wwei.yu (talk) 22:09, 28 March 2020 (EDT)Wei

1980 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 28	Followed by Problem 30
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1980 AHSME Problems/Problem 29

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