2025 AMC 10A Problems/Problem 3
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 
?
Solution 1: Casework
You can split the problem into two cases:
Case 
: The two sides are both smaller than , which means that they range from 
to 
. There are 
such cases.
Case 
: There are two sides of length , so the last side must be in the range to . There are such cases. Keep in mind, an equilateral triangle also counts as an isosceles triangle, since it has at least 2 sides.
Therefore, the total number of cases is 
~cw, minor edit by sd
Video Solution
~MK
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
See Also
| 2025 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
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| All AMC 10 Problems and Solutions | ||
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