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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