Found problems: 27
2023 AMC 12/AHSME, 17
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
$\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$
2023 AMC 12/AHSME, 15
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$