1. Compare the fractions $f_{1}=\frac{a+125^{725}}{a+625^{544}}$ and $f_{2}=\frac{b+121^{1007}}{b+343^{671}}$, where $a$ and $b$ are natural numbers.
Solution: $f_{1}=\frac{a+125^{725}}{a+625^{544}}=\frac{a+\left(5^{3}\right)^{725}}{a+\left(5^{4}\right)^{544}}=\frac{a+5^{2175}}{a+5^{2176}}$

- since $5^{2175}<5^{2176}$, we obtain that $a+5^{2175}<a+5^{2176}$, thus $f_{1}<1$.

For $f_{2}=\frac{b+11^{2014}}{b+7^{2013}}$:

- since $11^{2014}>7^{2013}$, we obtain that $b+11^{2014}>b+7^{2013}$, thus $f_{2}>1$.

From $f_{1}<1$ and $f_{2}>1$ we conclude that $f_{1}<f_{2}$. | $1 \mathrm{p}$ |
| Conclusion: $f_{1}<f_{2}$. | $1 \mathrm{p}$ |