Question

Define $\begin{aligned} P(x)={}&(x-1^2)(x-2^2)\\ &\cdots(x-100^2). \end{aligned}$ How many integers $n$ are there such that $P(n)\leq 0?$

$4900$

$4950$

$5000$

$5050$

$5100$

Answer: E

Concepts:polynomial inequality counting integers in a range

Difficulty rating: 1660

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Written solution:

The polynomial changes sign at each square $1^2,2^2,\ldots,100^2$ , and its leading coefficient is positive. Thus $P(n)\le0$ for integers in the intervals $[1^2,2^2]$ , $[3^2,4^2]$ , $\ldots$ , $[99^2,100^2]$ .

For odd $k$ , the interval $[k^2,(k+1)^2]$ contains $(k+1)^2-k^2+1=2k+2$ integers. Summing over odd $k=1,3,\ldots,99$ gives $2(1+3+\cdots+99)$ $+2\cdot50=5000+100$ $=5100$ . Thus, E is the correct answer.

2020 AMC 10A Problem 17

Answer: E

Difficulty rating: 1660

Video solution:

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Problem 17 in Other Years