2011 AIME I Problems/Problem 15
For some integer , the polynomial has the three integer roots , , and . Find .
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, .
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. ,
, , which is too small to get
, is not divisible by or or , we can clearly tell that is too much
Hence, , . , .