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.



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