10. It is known that the 3 sides of a triangle are consecutive positive integers and the largest angle is twice the smallest angle. Find the perimeter of this triangle.
10. Ans: 15
Let $\angle C=2 \angle A$ and $C D$ the bisector of $\angle C$. Let $B C=x-1, C A=x$ and $A B=x+1$. Then $\triangle A B C$ is similar to $\triangle C B D$. Thus $B D / B C=B C / A B$ so that $B D=(x-1)^{2} /(x+1)$. Also $C D / A C=C B / A B$ so that $A D=C D=$ $x(x-1) /(x+1)$.

As $A B=A D+B D$, we have $x(x-1) /(x+1)+(x-1)^{2} /(x+1)=x+1$. Solving this, the only positive solution is $x=5$. Thus the perimeter of the triangle is $4+5+6=15$