Let f(x) be a polynomial with integer coefficients such that f(5)=f(7)=20. What is the smallest possible value of |f(0)|?
+6252 consider g(x)=f(x)−20g(x)=a(x−5)(x−7)=a(x2−12x+35)and for integer coefficients it's clear that a∈Zg(0)=35af(0)=g(0)+20=35a+20|f(0)|=|35a+20|It should be fairly clear that |f(0)| is minimized by a=−1and thus |f(0)|=15 is the smallest possible value
There's holes in this. f isn't necessarily degree 2. I'll need to think about it a bit to make it more rigorous.
