2021 JMC 10 Problems/Problem 12

Problem

Mihir draws line $y=2x$ and Nathan draws line $x+y = n$ for an integer $n.$ The two lines divide the region $y \ge x^2$ into four regions, with regions possibly having infinite area. What is the sum of all possible values of $n?$

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 23 \qquad\textbf{(D) } 29 \qquad\textbf{(E) } 32$

Solution

The line $y=2x$ intersects the curve $y = x^2$ at two points, namely $(0,0)$ and $(2,4).$ Note that the line $x+y = n$ must intersect $y=2x$ at a point strictly between the points $(0,0)$ and $(2,4)$ to divide $y \ge x^2$ into four regions. If $x+y=n$ intersects $y=2x$ at $(0,0),$ we have $n=0.$ Similarly, if $x+y=n$ intersects $y=2x$ at $(2,4),$ we have $n=6.$ So, $0 < n <6.$ The sum of all possible $n$ is equal to $15.$

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2021 JMC 10 Problems/Problem 12

Problem

Solution