Problem 2. Find the number of the subsets $B$ of the set $\{1,2, \ldots, 2005\}$ having the following property: the sum of the elements of $B$ is congruent to 2006 modulo 2048 .

Emil Kolev
2. Let us consider the set $\left\{1,2,2^{2}, \ldots, 2^{10}\right\}$. Since every number from 0 to 2047 can be represented in a unique way as a sum of powers of 2 (elements of our set), we conclude that for every $i, 0 \leq i \leq 2047$, there is a unique subset of $\left\{1,2,2^{2}, \ldots, 2^{10}\right\}$ such that the sum of its elements is equal to $i$ ( 0 corresponds to the empty set).

We now consider a set $A$ with the following property: for every $i$ the number of the subsets of $A$ such that the sums of their elements are congruent to $i$ modulo 2048 does not depend on $i$. It is easy to see that for every $a$ the set $A \cup\{a\}$ has the same property. Since $\left\{1,2,2^{2}, \ldots, 2^{10}\right\} \subset\{1,2,3, \ldots, 2005\}$, we conclude that the number of the subsets $B$ of $\{1,2,3, \ldots, 2005\}$ such that the sums of their elements are congruent to $i$ modulo 2048 is equal to $\frac{2^{2005}}{2048}=$ $\frac{2^{2005}}{2^{11}}=2^{1994}$ and this is the required number.