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Difference between revisions of "2024 AMC 10B Problems/Problem 19"
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==Problem==
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==Solution 1==
 
==Solution 1==
 
Let's look at zero slope first. All lines of such form will be expressed in the form <math>y=k</math>, where <math>k</math> is some real number. If <math>k</math> is an integer, the line passes through infinitely many lattice points. One such example is <math>y=1</math>. If <math>k</math> is not an integer, the line passes through <math>0</math> lattice points. One such example is <math>y=1.1</math>. So we have <math>2</math> cases.
 
Let's look at zero slope first. All lines of such form will be expressed in the form <math>y=k</math>, where <math>k</math> is some real number. If <math>k</math> is an integer, the line passes through infinitely many lattice points. One such example is <math>y=1</math>. If <math>k</math> is not an integer, the line passes through <math>0</math> lattice points. One such example is <math>y=1.1</math>. So we have <math>2</math> cases.

Revision as of 01:31, 14 November 2024

Problem

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

See also

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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