## Subject I

Let the expression $A=53 \cdot\left(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{2013 \cdot 2014}\right)$.

a) Calculate the value of the expression $A$.

b) Prove that $A$ can be written in the form $\frac{a \cdot b \cdot c}{d \cdot e}$, where the sum of the digits of the numbers $a, b$, and $c$ is equal to the sum of the digits of the numbers $d$ and $e$.
## SUBJECT I

Let the expression $A=53 \cdot\left(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{2013 \cdot 2014}\right)$.

a) Calculate the value of the expression $A$.

b) Prove that $A$ can be written in the form $\frac{a \cdot b \cdot c}{d \cdot e}$, where the sum of the digits of the numbers $a, b$, and $c$ is equal to the sum of the digits of the numbers $d$ and $e$.

## Solution

a) $A=53 \cdot\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdots+\frac{1}{2013}-\frac{1}{2014}\right)=53 \cdot\left(1-\frac{1}{2014}\right)=53 \cdot \frac{2013}{2014}=\frac{2013}{38}$
b) $A=\frac{2013}{38}=\frac{3 \cdot 11 \cdot 61}{2 \cdot 19}$,

The sum of the digits in the numerator and denominator is equal to 12.