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
Solution 1
Adding the above two equations gives , and so we can deduce that
.
Similarly, plugging in and
gives
and
. Now,
Therefore
and
. Finally,
So,
, or equivalently
.
Substituting these equations into the original polynomial , we find that at
,
.
The remaining coefficients
and
are now completely arbitrary because the original equations impose no more restrictions on them. Hence, for the final equation to hold for all possible
, we must have
.
As the answer format implies that the -coordinate of the root is non-integral,
. The format also implies that
is positive, so
. Substituting
into
and reducing to a quadratic yields
, in which the only non-integral root is
, so
.
The answer is .
Solution 2
Consider the cross section of on the plane
. We realize that we could construct the lines/curves in the cross section such that their equations multiply to match the form of
(same degree of
and
in terms) and they include the eight given points. One simple way to do this would be to use the equations
,
, and
, giving us
.
Another way to do this would to use the line and the ellipse,
. This would give
. (But
(Another way would be to use the hyperbola from Solution 1. Interesting that different curves both work.)
At this point, we consider that and
both must have
as a zero. A quick graph of the 4 lines and the ellipse used to create
and
gives nine intersection points. Eight of them are the given ones, and the ninth is
. The last intersection point can be found by finding the intersection points of
and
.
Finally, just add the values of
,
, and
to get
Solution 3 (Ansatz)
We can plug in the values to obtain
Now, this means that
After some simplifying, we obtain
Since ,
and we suspect that:
and
$$ (Error compiling LaTeX. Unknown error_msg)\frac{x-x^3}{xy(1-x)}=\frac{3}{2}\Leftrightarrow\frac{1+x}{y}=\frac{3}{2}$$<cmath>Leftrightarrow\frac{2}{3}+\frac{2x}{3}=y</cmath>.
Plugging this into the first equation, and factoring, and cancelling$ (Error compiling LaTeX. Unknown error_msg)(x+1)19x^2 -43x+10=0
(x,y)=\left(\frac{5}{19},\frac{16}{19}\right)\Longrightarrow a+b+c=5+16+19=\boxed{40}.$$ (Error compiling LaTeX. Unknown error_msg)\blacksquare$
~~pinkpig
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 |
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