Problem

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?

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

Solution 2=

0 slope: 0 points (e.g. y=1.5), moer than 2 (inf, e.g. y=1)

nonzero: 0 (eg y=2/3 x + 1/2), more than 2 (eg y=2x)

irrational: 0 (eg y =sqrt3*x+1) exactly 1 (eg y=sqrt3*x, which has an intersection at 0,0)

Hence a total of 6. ~mathboy282

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/c6nhclB5V1w?feature=shared

~ Pi Academy

2024 AMC 10B (Problems • Answer Key • Resources)
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2024 AMC 10B Problems/Problem 19

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Solution 1

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Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

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