2024 AMC 12B Problem 22

When $\angle B = 2\angle A,$ the side lengths satisfy $b^2 = a(a + c),$ where $a = BC,$ $b = CA,$ $c = AB.$ So $c = \dfrac{b^2 - a^2}{a}$ must be a positive integer, and the sides must form a valid triangle. Also $b\gt a,$ because $\angle B=2\angle A\gt\angle A.$ The perimeter is $a+b+c=b+\dfrac{b^2}{a}\gt2b.$ Therefore a perimeter below $15$ would require $b\le6.$ For $b=2,3,4,5,6,$ the divisors $a\lt b$ of $b^2$ give the possible pairs $$\begin{gathered} (b,a)=(2,1),(3,1),\\ (4,1),(4,2),(5,1),\\ (6,1),(6,2),(6,3),(6,4). \end{gathered}$$ Substitution gives a degenerate or invalid triangle in every case except $(a,b)=(4,6),$ which gives $c=5.$ Thus $(4,5,6)$ is the first valid triangle, and its perimeter is $15.$ Thus, the correct answer is C.