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 p
p
k^4 \equiv 1 \pmod(p)
k
p=2, 3
5$.
When$ (Error compiling LaTeX. Unknown error_msg)p=2r
ra
rb
rc
rd$are all 1 mod 2, or the sum wouldn't be 2. Any pairwise sum is 2.
When$ (Error compiling LaTeX. Unknown error_msg)p=3r
1
2$. Thus, four pairwise sums sum to 3.
When$ (Error compiling LaTeX. Unknown error_msg)p=5r
a, b, c, d
1, 2, 3
4$, 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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