## Problem 1

Calculate the sum

$$
S=[\lg 1]+[\lg 2]+[\lg 3]+\cdots+\left[\lg 10^{2014}\right]
$$
## Solution.

$$
\text { If } p \in \mathbb{N} \text {, then }[\lg k]=p \Leftrightarrow p \leq \lg k<p+1 \ldots \ldots 1 p
$$

The sum is equal to $S=9 \cdot \sum_{p=0}^{2013} p \cdot 10^{p}+2014$......3p

which in the end is

$S=\frac{10}{9}\left(2013 \cdot 10^{2014}-2014 \cdot 10^{2013}+1\right)+2014$