Difference between revisions of "2011 AIME I Problems/Problem 15"
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] |
Revision as of 07:49, 29 March 2011
Problem
For some integer , the polynomial
has the three integer roots
,
, and
. Find
.
Solution
With Vieta's formula, we know that , and
.
since any one being zero will make the the other 2
.
. WLOG, let
.
Then if , then
and if
,
.
We know that ,
have the same sign. So
. (
and
)
Also, maximize when
if we fixed
. Hence,
.
So .
so
.
Now we have limited a to .
Let's us analyze .
Here is a table:
![]() | ![]() |
---|---|
![]() | ![]() |
![]() | ![]() |
![]() | ![]() |
![]() | ![]() |
![]() | ![]() |
We can tell we don't need to bother with ,
, So
won't work.
,
is not divisible by
,
, which is too small to get
,
is not divisible by
or
or
, we can clearly tell that
is too much
Hence, ,
.
,
.
Answer:
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by - | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |