# Difference between revisions of "1984 AIME Problems/Problem 12"

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== Solution == | == Solution == | ||

− | {{ | + | If <math>\displaystyle f(2+x)=f(2-x)</math>, then substituting <math>\displaystyle t=2+x</math> gives <math>\displaystyle f(t)=f(4-t)</math>. Similarly, <math>\displaystyle f(t)=f(14-t)</math>. In particular, |

+ | |||

+ | <math>\displaystyle f(t)=f(14-t)=f(14-(4-t))=f(t+10)</math> | ||

+ | |||

+ | |||

+ | 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 | ||

+ | |||

+ | <math>\sin \frac{\pi x}{10}\sin \frac{\pi (x-4)}{10}</math> | ||

+ | |||

+ | satisfies the conditions and has no other roots. | ||

+ | |||

+ | |||

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

+ | |||

+ | <math>401</math> | ||

== See also == | == See also == | ||

* [[1984 AIME Problems/Problem 11 | Previous problem]] | * [[1984 AIME Problems/Problem 11 | Previous problem]] | ||

* [[1984 AIME Problems/Problem 13 | Next problem]] | * [[1984 AIME Problems/Problem 13 | Next problem]] | ||

* [[1984 AIME Problems]] | * [[1984 AIME Problems]] |

## Revision as of 19:58, 26 March 2007

## Problem

A function is defined for all real numbers and satisfies and for all . If is a root for , what is the least number of roots must have in the interval ?

## Solution

If , then substituting gives . Similarly, . In particular,

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

satisfies the conditions and has no other roots.

In the interval , there are 201 multiples of 10 and 200 numbers that are congruent to 4 modulo 10, therefore the minimum number of roots is