tmua.me Extreme Set C Paper 2 · Question 9

Let $p$ and $q$ be relatively prime positive integers, and let $b,c,d$ be rational numbers. Define

P(x)=\frac pq x^3+bx^2+cx+d.

Consider the following statement.

There are infinitely many positive integers $n$ for which

\frac{P(n)}{n^2+1}

is an integer.

Which of the following correctly describes the condition that $c=p/q$ , $d=b$ , and $pb$ is an integer, in relation to this statement?

AThe condition is necessary but not sufficient for the statement.

BThe condition is sufficient but not necessary for the statement.

CThe condition is both necessary and sufficient for the statement.

DThe condition is neither necessary nor sufficient for the statement.

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