Let P(x) be a polynomial with integer coefficients. Suppose that for three distinct integers a,b,c we have P(a)=P(b)=P(c)=-5, and that P(1)=6. What is the maximum possible value of a2+b2+c2?
Let's create a new polynomial q(x)=p(x-1)+5
Then q has zeros a'=a+1, b'=b+1, c'=c+1 and q(0)=11
q(x)=d(x-a')(x-b')(x-c') for some integer d.
q(0)=da'b'c'=11
11 is prime and since a',b',c' must be distinct we can't let d=11.
The only two choices are
(a',b',c') = (1,-1,11) in some order and d'=-1
or
(a',b',c') = (1,-1,-11) in some order and d'=1
The first choice makes (a,b,c)=(0,-2,10) and a^2+b^2+c^2=104
The second makes (a,b,c)=(0,-2,-12) and a^2+b^2+c^2=148
The smaller is the first so the answer is 104
For completeness the polynomial is
p(x) = -x(x-2)(x+10)-5 = -x^3-8x^2+20x-5
https://www.desmos.com/calculator/tvxnbgyknx
|
|
Posted by Jer
on 2025-01-28 13:02:51 |
possible solution
