2019 AIME I Problems/Problem 10
The 2019 AIME I takes place on March 13, 2019.
Contents
Problem 10
For distinct complex numbers , the polynomial
can be expressed as
, where
is a polynomial with complex coefficients and with degree at most
. The value of
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Solution
In order to begin this problem, we must first understand what it is asking for. The notation
simply asks for the absolute value of the sum of the product of the distinct unique roots of the polynomial taken two at a time or
Call this sum
.
Now we can begin the problem. Rewrite the polynomial as . Then we have that the roots of
are
.
By Vieta's formulas, we have that the sum of the roots of is
. Thus,
Similarly, we also have that the the sum of the roots of taken two at a time is
This is equal to
Now we need to find and expression for in terms of
. We note that
Thus,
.
Plugging this into our other Vieta equation, we have . This gives
. Since 343 is relatively prime to 9,
.
Solution 2
This is a quick fake solve using where
and only
.
By Vieta's, and
Rearranging gives
and
giving
.
Substituting gives which simplifies to
,
,
,
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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