1957 AHSME Problems/Problem 43

Problem

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $x$ -axis, the line $x = 4$ , and the parabola $y = x^2$ is:

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \infty$

Solution

$[asy] path p = (0,0){right}..(1,1)..(2,4)..(3,9)..(4,16); // Shaded Region fill(p--(4,0)--cycle,lightred); // x-Axis draw((-4,0)--(16,0), arrow=Arrows); label("$x$",(18,0)); // y-Axis draw((0,-4)--(0,16), arrow=Arrows); label("$y$",(0,18)); // y=x^2 draw(p); // x=4 draw((4,-5)--(4,20), arrow=Arrows(TeXHead)); [/asy]$

We want to find the number of lattice points in and on the boundary of the shaded region in the diagram. To do this, we will look at the integer values of $x$ from $0$ to $4$ . At a given value of $x$ , the amount of lattice points in the region is $x^2+1$ , because all of the integers from $0$ up to and including $x^2$ are in the region. Thus, evaluating this expression at at $x=0,1,2,3,$ and $4$ and adding the results together, we see that the number of lattice points is $1+2+5+10+17=35$ , so our answer is $\boxed{\textbf{(B) }35}$ .

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 42	Followed by Problem 44
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1957 AHSME Problems/Problem 43

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