2024 AMC 10B Problems/Problem 19
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"?
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Solution 1
Let's look at zero slope first. All lines of such form will be expressed in the form 
, where 
is some real number. If is an integer, the line passes through infinitely many lattice points. One such example is 
. If is not an integer, the line passes through 
lattice points. One such example is 
. So we have 
cases.
Let's now look at the second case. The line has slope 
, where 
and 
are relatively prime integers. The line has the form 
. If the line passes through lattice point 
, then the line must also pass through the lattice point 
, where 
are all integers. Therefore, the line can pass through infinitely many lattice points but it cannot pass through exactly 
or . The line can pass through lattice points, such as 
. This contributes more cases.
If the line has an irrational slope, it can never pass through more than lattice points. We prove this using contradiction. Let's say the line passes through lattice points and 
. Then the line has slope 
, which is rational. However, the slope of the line is irrational. Therefore, the line can pass through at most lattice point. One example of this is 
. This line contributes final cases.
Our answer is therefore 
.
~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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