1997 AHSME Problems/Problem 3

Problem 3

If $x$ , $y$ , and $z$ are real numbers such that

$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$ ,

then $x + y + z =$

$\mathrm{(A)\ } -12 \qquad \mathrm{(B) \ }0 \qquad \mathrm{(C) \ } 8 \qquad \mathrm{(D) \ } 12 \qquad \mathrm{(E) \ }50$

Solution

If the sum of three squared expressions is zero, then each expression itself must be zero, since $a^2 \ge 0$ with the equality iff $a=0$ .

In this case, $x-3=0$ , $y-4=0$ , and $z-5=0$ . Adding the three equations and moving the constant to the right gives $x + y + z = 12$ , and the answer is $\boxed{D}$ .

1997 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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1997 AHSME Problems/Problem 3

Problem 3

Solution

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