1984 AIME Problems/Problem 12

Problem

A function $\displaystyle f$ is defined for all real numbers and satisfies $\displaystyle f(2+x)=f(2-x)$ and $\displaystyle f(7+x)=f(7-x)$ for all $\displaystyle x$ . If $\displaystyle x=0$ is a root for $\displaystyle f(x)=0$ , what is the least number of roots $\displaystyle f(x)=0$ must have in the interval $\displaystyle -1000\leq x \leq 1000$ ?

Solution

If $\displaystyle f(2+x)=f(2-x)$ , then substituting $\displaystyle t=2+x$ gives $\displaystyle f(t)=f(4-t)$ . Similarly, $\displaystyle f(t)=f(14-t)$ . In particular,

$\displaystyle f(t)=f(14-t)=f(14-(4-t))=f(t+10)$

Since 0 is a root, all multiples of 10 are roots, and anything of the form "4 minus a multiple of 10" (that is, anything congruent to 4 modulo 10) are also roots. To see that these may be the only integer roots, observe that the function

$\sin \frac{\pi x}{10}\sin \frac{\pi (x-4)}{10}$

satisfies the conditions and has no other roots.

In the interval $-1000\leq x\leq 1000$ , there are 201 multiples of 10 and 200 numbers that are congruent to 4 modulo 10, therefore the minimum number of roots is

1984 AIME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1984 AIME Problems/Problem 12

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