National Olympiad of Argentina

Write $v=\frac{2uw}{u+w}$ as $\frac{1}{v}=\frac{u+w}{2uw}$. This gives $\frac{1}{v}=\frac{1}{2}\left(\frac{1}{u}+\frac{1}{w}\right)$, or $\frac{1}{v}-\frac{1}{u}=\frac{1}{w}-\frac{1}{v}$. So look at the sequence of reciprocals of the given numbers: $\frac{1}{v}-\frac{1}{u}=\frac{1}{w}-\frac{1}{v}$ means that consecutive reciprocals differ by the same amount which we denote by $d$.

Hence the last reciprocal $\frac{1}{1/101}=101$ can be obtained from the first one $\frac{1}{1/100}=100$ by adding $d$ 20 times. Thus $101=100+20d$, yielding $d=\frac{1}{20}$. To obtain the 15th reciprocal we add $14d$ to the first one, 100, which gives $100+14\cdot \frac{1}{20}=\frac{1007}{10}$. Therefore the 15th original number is $\frac{10}{1007}$.