2008 AIME I Problems/Problem 13
Problem
Let
Suppose that
There is a point for which
for all such polynomials, where
,
, and
are positive integers,
and
are relatively prime, and
. Find
.
Solution
Adding the above two equations gives , and so we can deduce that
.
Similarly, plugging in and
gives
and
. Now,
Therefore and
. Finally
So
.
Now
.
In order for the above to be zero, we must have
and
Canceling terms on the second equation gives us
. Plugging that into the first equation and solving yields
, and
.
See also
2008 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |