5. For a non-empty subset $X$ of the set $\{1,2,3, \ldots, 42\}$, denote by $v(X)$ the sum of the elements of the set $X$. (For example, $v(\{1,3,8\})=12$.)

Calculate the sum of all numbers $v(X)$, as $X$ ranges over all non-empty subsets of the set $\{1,2,3, \ldots, 42\}$.

Solve the problem independently. You have 150 minutes available for solving.

The use of notes, literature, or a pocket calculator is not allowed.

# Solutions to the selection competition problems


#### Abstract

Each problem is worth 7 points. When evaluating each problem, reasonably consider the provided scoring guide. A competitor should not receive more than 3 points for a problem if the partial solution does not clearly lead to the final solution of the problem.
IV /5. Let the sought sum be denoted by $V$. Each of the numbers from 1 to 42 appears in the sum $V$ as many times as there are subsets of the set $\{1,2,3, \ldots, 42\}$ that contain this number. There are $2^{41}$ such subsets, so

$$
S=(1+2+3+\ldots+42) 2^{41}=21 \cdot 43 \cdot 2^{42}
$$

The realization that each number from 1 to 42 appears in the sum $2^{41}$ times 5 points Calculation of $(1+2+3+\ldots+42) 2^{41}=21 \cdot 43 \cdot 2^{42}$ 2 points