Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that$$f(p) = f(q) = f(r) = f(s) = 5.$$If $t$ is an integer and $f(t)>5$, what is the smallest possible value of $f(t)$?

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michaelcai Jan 30, 2018

#1**0 **

Rewritten:

Let f(x) be a polynomial with integer coefficients.

Suppose there are four distinct integers p,q,r,s such that f(p) = f(q) = f(r) = f(s) = 5. (Can there be more??)

If t is an integer and f(t)>5,

what is the smallest possible value of f(t) ?

I'd like to see this one done too, it makes little sense to me .

Melody Jan 30, 2018

#2**+3 **

Conversation between friend and me:

So you can write f(x) = (x-p)(x-q)(x-r)(x-s)+5, which will satisfy the thing and then minimize it since everything is integer

oh yeah

but how to minimize it]

isnt the minimumn 5?

f(x)>5 though

Plug in x=t

ok

Basically minimize positive value of four different integers (t-whatever)

Value of product of four

f(t) = (t-p)(t-q)(t-r)(t-s)+5>5, so (t-p)(t-q)(t-r)(t-s)>0

?

(t-p)(t-q)(t-r)(t-s) would be 1*2*3*4?

at minimum

Yeah so the product of the four t's need to be positive and distinct since p,q,r,s are different

so the answer wold be 24?

No

Wait

The product has to be positive but not each term

oh

And add 5

-1, 1, -2, 2 product=4, plus 5 is 9?

so it would be 9?

And the answer is 9

michaelcai Feb 2, 2018