A not so easy quadratic equation problem for RMO or IITJEE maths

Question:

Suppose that we are given a monic quadratic polynomial p(t). Prove that for any integer n, there exists an integer k such that p(n)p(n+1)=p(k).

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Solution:

Let p(t)=t^{2}+bt+c and which implies p(t+1)=(t+1)^{2}+b(t+1)+c. We want p(n)p(n+1)=p(k). By trial and error, we get k=t(t+1)+bt+c.

By the way, we could have gotten the same solution by method of undetermined coefficients. But that would also need intelligent guess-works.

Nalin Pithwa

PS: I will post the solution after some time. Meanwhile, please try.

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