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2025 AMC 10A Problems/Problem 3: Difference between revisions

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==Solution 1: Casework==
==Solution 1: Casework==
Split the problem into two cases:<br>
You can split the problem into two cases:<br>
Case <imath>1</imath>: The two sides are both smaller than <imath>2025</imath>, which means that they range from <imath>1013</imath> to <imath>2024</imath>. There are <imath>1012</imath> such cases.<br>
Case <imath>1</imath>: The two sides are both smaller than <imath>2025</imath>, which means that they range from <imath>1013</imath> to <imath>2024</imath>. There are <imath>1012</imath> such cases.<br>
Case <imath>2</imath>: There are two sides of length <imath>2025</imath>, so the last side must be in the range <imath>1</imath> to <imath>2025</imath>. There are <imath>2025</imath> such cases. Keep in mind, an equilateral triangle also counts as an isosceles triangle, since it has *at least* 2 sides.<br>
Case <imath>2</imath>: There are two sides of length <imath>2025</imath>, so the last side must be in the range <imath>1</imath> to <imath>2025</imath>. There are <imath>2025</imath> such cases. Keep in mind, an equilateral triangle also counts as an isosceles triangle, since it has <b>at least</b> 2 sides.<br>
Therefore, the total number of cases is <imath>1012 + 2025 = \boxed{3037}</imath><br>
Therefore, the total number of cases is <imath>1012 + 2025 = \boxed{3037}</imath><br>
-cw
~cw, minor edit by sd

Revision as of 15:12, 6 November 2025

How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$?

$\textbf{(A)}~2025\qquad\textbf{(B)}~2026\qquad\textbf{(C)}~3012\qquad\textbf{(D)}~3037\qquad\textbf{(E)}~4050$

Solution 1: Casework

You can split the problem into two cases:
Case $1$: The two sides are both smaller than $2025$, which means that they range from $1013$ to $2024$. There are $1012$ such cases.
Case $2$: There are two sides of length $2025$, so the last side must be in the range $1$ to $2025$. There are $2025$ 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 $1012 + 2025 = \boxed{3037}$
~cw, minor edit by sd

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