Subject 1. Let the sum $S=101+1001+10001+\ldots+\underbrace{100 \ldots 001}_{2015 \text { digits }}$.

a. Find how many digits the middle term of the sum has.

b. Find how many zeros are used to write all the terms of the sum.

c. Calculate the sum.
Subject 1.

| a. | The sum has 2013 terms, so the middle term is the 1007th term and has 1009 digits | $\mathbf{2 p}$ |
| :--- | :--- | :--- |
| b. | The first term has one 0, the second has two 0s, ..., the last has 2013 zeros | $\mathbf{2 p}$ |
|  | $1+2+\ldots+2013=2027091$ zeros are used | $\mathbf{3 p}$ |
| c. | $S=100+1000+\ldots+\underbrace{100 \ldots 00}_{\text {2015 digits }}+2013=\underbrace{111 \ldots 100}_{\text {2015 digits }}+2013=\underbrace{111 \ldots 13113}_{\text {2011 digits }}$ |  |