51st Ukrainian National Mathematical Olympiad, 4th Round

Let $N$ be the number of twin numbers that do not exceed $M$. Then $N\ge N_3+N_4-N_{12}$, since all the numbers from $M_3$ and $M_4$ are twin, as they have divisors $1,3$ and $2,4$ respectively, and $M_2$ consists of all the numbers that belong to both $M_3$ and $M_4$. But there are also twin numbers that do not exceed $M$ and do not belong neither to $M_3$ nor to $M_4$, for example $35=5\cdot 7$, and so we even have that $N>N_3+N_4-N_{12}$. Since $M$ is divisible by $12$, we obtain: $$N>\frac{M}{3}+\frac{M}{4}-\frac{M}{12}=\frac{M}{2},$$ which precisely means that there are more twin numbers than the numbers that are not twin among the first $M=20112012$ natural numbers.