1. Citizen Sidorov has exactly as much money as needed to buy a ton of roundels and a ton of smuggles. If he buys 20% more roundels, he will get a 40% discount on smuggles, and the remaining money will be enough to buy a ton of smuggles. If he buys 40% more smuggles, he will get a 20% discount on roundels, and the remaining money will be enough to buy a ton of roundels. Which is more expensive and by how many times: a ton of roundels or a ton of smuggles? (In both cases, not all the money necessarily has to be spent)
Answer. A ton of kruglik costs twice as much as a ton of shmuglik. Solution. Let a ton of kruglik cost $x$ money, and a ton of shmuglik $y$ money. Then citizen Sidorov has ( $x+y$ ) money. If he buys 20% more kruglik, and a ton of shmuglik with a 40% discount, he will spend $1.2 x+0.6 y$ and this is no more than ( $x+y$ ). If he buys 40% more shmuglik, and a ton of kruglik with a 20% discount, it will cost $1.4 y+0.8 x$, and this is no more than $(x+y)$. We get two inequalities: $\left\{\begin{array}{l}1.2 x+0.6 y \leq x+y \\ 0.8 x+1.4 y \leq x+y\end{array}\right.$. From the first inequality it follows that $0.2 x \leq 0.4 y$, i.e., $x \leq 2 y$. And from the second, it follows that $0.4 y \leq 0.2 x$, i.e., $2 y \leq x$. Comparing the two inequalities, we conclude that $x=2 y$.