Dạng toán liên phân số quen thuộc trong APMOPS 2013

Question:

If we write $\displaystyle\frac{2013}{1990}$ in the form $\displaystyle a+\frac{1}{b+\displaystyle\frac{1}{c+\displaystyle\frac{1}{d+\displaystyle\frac{1}{e}}}},$

where $a,b,c,d,e$ are positive integers, what is the value of $a+b+c+d+e$?

Solution:

We have known that $\displaystyle\frac{2013}{1990}$ can be written in the unique form:

$\displaystyle\frac{2013}{1990}=1+\frac{23}{1990}=1+\frac{1}{\displaystyle\frac{1990}{23}}$

$\displaystyle=1+\frac{1}{86+\displaystyle\frac{12}{23}}=1+\displaystyle\frac{1}{86+\displaystyle\frac{1}{\displaystyle\frac{23}{12}}}$

$\displaystyle=1+\frac{1}{86+\displaystyle \frac{1}{1+\displaystyle\frac{11}{12}}}=1+\frac{1}{86+\displaystyle \frac{1}{1+\displaystyle\frac{1}{\displaystyle\frac{12}{11}}}}$

$\displaystyle=1+\frac{1}{86+\displaystyle \frac{1}{1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{11}}}}.$

Hence $a+b+c+d+e=1+86+1+1+11=100.$