1997 USAMO Problems/Problem 1
Let be the prime numbers listed in increasing order, and let be a real number between and . For positive integer , define
where denotes the fractional part of . (The fractional part of is given by where is the greatest integer less than or equal to .) Find, with proof, all satisfying for which the sequence eventually becomes .
All rational numbers between 0 and 1 will eventually become 0 under this iterative process. To begin, note that by definition, all rational numbers can be written as a quotient of coprime integers. Let , where are coprime positive integers. Since , . Now From this, we can see that applying the iterative process will decrease the value of the denominator, since . Moreover, the numerator is always smaller than the denominator, thanks to the fractional part operator. So we have a strictly decreasing denominator that bounds the numerator. Thus, the numerator will eventually become 0.
On the other hand, irrational will never become 0, because will always be irrational.
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