2024 AMC 10B Problems/Problem 19

Solution 1

Let's look at zero slope first. All lines of such form will be expressed in the form $y=k$ , where $k$ is some real number. If $k$ is an integer, the line passes through infinitely many lattice points. One such example is $y=1$ . If $k$ is not an integer, the line passes through $0$ lattice points. One such example is $y=1.1$ . So we have $2$ cases.

Let's now look at the second case. The line has slope $\frac{p}{q}$ , where $p$ and $q$ are relatively prime integers. The line has the form $y = \frac{p}{q}x + m$ . If the line passes through lattice point $(a,b)$ , then the line must also pass through the lattice point $(a+kq, b+kp)$ , where $a,b,k$ are all integers. Therefore, the line can pass through infinitely many lattice points but it cannot pass through exactly $1$ or $2$ . The line can pass through $0$ lattice points, such as $y=x+0.5$ . This contributes $2$ more cases.

If the line has an irrational slope, it can never pass through more than $2$ lattice points. We prove this using contradiction. Let's say the line passes through lattice points $(a,b)$ and $(c,d)$ . Then the line has slope $\frac{d-b}{c-a}$ , which is rational. However, the slope of the line is irrational. Therefore, the line can pass through at most $1$ lattice point. One example of this is $y=\sqrt{2}x$ . This line contributes $2$ final cases.

Our answer is therefore $\boxed{6}$ .

~lprado

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2024 AMC 10B Problems/Problem 19

Solution 1