2024/25 AHSMC Part 1 Problem 13

Problem 13 of 16

The number of points $(x,y)$ with integer coordinates $x$ and $y$ , which are inside the circle $x^2 + y^2 = 1000$ , and such that $|3x+1| + |x-1| = y$ is:

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Solution

If $x \ge 1$ then $y = 4x$ . Writing $x = k \ge 1$ we need $17k^2 < 1000$ , so $k \le 7$ : seven points $(k,4k)$ .

If $-\dfrac{1}{3} \le x < 1$ then $x = 0$ and $y = 2$ , one interior point.

If $x < -\dfrac{1}{3}$ then $y = -4x$ . Writing $x = -k$ with $k \ge 1$ again gives $k \le 7$ : seven points $(-k,4k)$ .

In all, $15$ points. The answer is (D).

University of Alberta

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