1997 USAMO Problems/Problem 3
Problem
Prove that for any integer , there exists a unique polynomial
with coefficients in
such that
.
Solution
The problem implies that . We define
. Then, after expanding, the coefficient of
is
where
. The constant term is
. Since this has to be within the set
,
must be unique according to
. The linear term is determined by
, which also must be within the set
. Since
is determined uniquely by
, in order to keep the linear coefficient in the set
,
is also determined uniquely. With
and
determined uniquely, we move on to the general coefficient.
Note that is in the set
. We define
to be a function determining the remainder when
is divided by
. Thus, for all
,
. Note also that
, since
J_h(x)
J_h(7a_{k-1} + a_{k-2}) = 10a_k + 7a_{k-1} + a_{k-2}
for all
. Thus,
is determined uniquely, since all coefficients
where
are uniquely determined, which in turn determines a unique polynomial
.
See Also
1997 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.