1999 USAMO Problems/Problem 3
Problem
Let be a prime and let be integers not divisible by , such that for any integer not divisible by . Prove that at least two of the numbers , , , , , are divisible by . (Note: denotes the fractional part of .)
Solution
We see that means that . Now, since does nto divide and is prime, their GCD is 1 so .
Since , then we see that they have to represent mods , and thus, our possible values of are all such that for all . This happens when or .
When then is odd, meaning , , and are all 1 mod 2, or the sum wouldn't be 2. Any pairwise sum is 2.
When then is not divisible by 3, thus two are , and the other two are . Thus, four pairwise sums sum to 3.
When then is not divisible by 5 so are and , so two pairwise sums sum to 5.
All three possible cases work so we are done.
See Also
1999 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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