1967 AHSME Problems/Problem 31

Problem

Let $D=a^2+b^2+c^2$ , where $a$ , $b$ , are consecutive integers and $c=ab$ . Then $\sqrt{D}$ is:

$\textbf{(A)}\ \text{always an even integer}\qquad \textbf{(B)}\ \text{sometimes an odd integer, sometimes not}\\ \textbf{(C)}\ \text{always an odd integer}\qquad \textbf{(D)}\ \text{sometimes rational, sometimes not}\\ \textbf{(E)}\ \text{always irrational}$

Solution

Let $a=x, b=x+1, c = x(x+1)$ . Then $D = x^2 + (x+1)^2 + x^2(x+1)^2$ , which simplifies to $x^4 + 2x^3 + 3x^2 + 2x + 1$ .

From the options, we want to test if $D$ is always a perfect square. Because the polynomial expression for $D$ is quartic, if it is a perfect square, it would be the square of a quadratic expression. Thus, $D$ could be written in the form $(Ax^2 + Bx + C)^2$ for some $(A, B, C)$ .

Setting $(Ax^2 + Bx + C)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$ , we can compare coefficients. From the $x^4$ coefficient, we get $A = \pm 1$ . Note that if $(Ax^2 + Bx^2 + C)^2$ works, so does $(-Ax^2 - Bx - C)^2$ , so we can arbitrarily pick $A=1$ .

We now have $(x^2 + Bx + C)^2= x^4 + 2x^3 + 3x^2 + 2x + 1$ . Setting the cubic terms equal gives $2B = 2$ , or $B = 1$ . This leaves $(x^2 + x + C)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$ . We can quickly inspect the constant term to determine that $C = \pm 1$ . We reject $C = -1$ , since the quadratic and linear terms won't match up, which leaves $(x^2 + x + 1)^2$ as the only possibility - and, in fact, it works.

Thus, $D$ is always the square of an integer - namely $x^2 + x + 1$ . This in turn means that $\sqrt{D}$ is always rational, which leaves choices $A, B, C$ as the only possible correct answers.

The question now is whether $\sqrt{D}$ , or $x^2 + x + 1$ , is odd, even, or could be both. We have two cases for $x$ :

If $x \equiv 0 \pmod 2$ , then $x^2 \equiv 0^2 \pmod 2$ . This means $x^2 + x + 1 \equiv 0 + 0 + 1 \equiv 1 \pmod 2$ .

If $x \equiv 1 \pmod 2$ , then $x^2 \equiv 1^2 \pmod 2$ , and $x^2 + x + 1 \equiv 1 + 1 + 1 \equiv 1 \pmod 2$ .

Either way, $x^2 + x + 1$ is an odd integer, and the answer is $\fbox{C}$

1967 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 30	Followed by Problem 32
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1967 AHSME Problems/Problem 31

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