9. Determine the number of acute-angled triangles (i.e., all angles are less than $90^{\circ}$ ) in which all angles (in degrees) are positive integers and the largest angle is three times the smallest angle.
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
9. Answer: (B)
Let the angles of the acute-angled triangle be $x^{*}, y^{*}, 3 x^{*}$, where the smallest angle is $x^{\circ}$. Then we have $x+y+3 x=180$ and $0<x \leq y \leq 3 x<90$. From the inequalities $x \leq y \leq 3 x$, we obtain $5 x \leq x+y+3 x \leq 7 x$, and hence it follows from the first equation that $\frac{180}{7} \leq x \leq 36$. Since $x$ is an integer and $3 x<90$, we deduce that $x=26,27,28,29$. Hence there are 4 acute-angled triangles whose angles are respectively $\left(26^{\circ}, 76^{\circ}, 78^{\circ}\right),\left(27^{\circ}, 72^{\circ}, 81^{\circ}\right),\left(28^{\circ}, 68^{\circ}, 84^{\circ}\right)$ and $\left(29^{\circ}, 64^{\circ}, 87^{\circ}\right)$.