Here's an ideia: Given a binary number $t$, define the position of its digits the order of the digit, for example:$t=1010001111010$, the positions 2,4,5,6,7,11,13 there is digit 1 and in the positions 1,3,8,9,10,12,14,15,16,17,18,... there is digit 0. Let $h=\lfloor{log_{2}k}\rfloor$. Let's color the positions using $(h+1)$ colors by the following ruler: If $2^{i}$ is the highest power of $2$(less than or equal to $2^{h}$) that divides $r$, then the position $r$ has the color $i$. So, a number $t$ will be loser iff for each $i$(0≤i≤h), the number of $1$s on positions of color $i$ is even. It can be proved by induction on $t$. The rest is easy.

Am I missing something or is the problem wrong for k=1? It claims that all the numbers less than 2^n are loser numbers which is clear absurd...

The problem is indeed wrong. If I remember corretly, it should say \(2^{N-\lfloor \frac{log(min\{N,k\}) - 1}{log 2} \rfloor}\)

In the original file there is nothing about corrections.

Johann Peter Dirichlet wrote: Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a winner number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it loser. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\) The problem is still wrong for $N=2$?

The correct expression, the number of non-negative loser integers, should be $2^{N-\lfloor \frac{\log(\min\{N,k\})}{\log 2} \rfloor - 1}$.

Anyone have the solution with the correct expoent?