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 \{ \frac{ra}p \}+\{ \frac{rb}p \}+\{ \frac{rc}p \}+\{ \frac{rd}p \} =2\mod ppk^4 \equiv 1 \pmod(p)kp=2, 35$.
When$ (Error compiling LaTeX. ! Missing $ inserted.)p=2rrarbrcrd$are all 1 mod 2, or the sum wouldn't be 2. Any pairwise sum is 2.
When$ (Error compiling LaTeX. ! Missing $ inserted.)p=3r12$. Thus, four pairwise sums sum to 3.
When$ (Error compiling LaTeX. ! Missing $ inserted.)p=5ra, b, c, d1, 2, 34$, 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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