1. In an online store, two types of New Year's gifts are sold. The first type of gift contains a toy, 3 chocolate, and 15 caramel candies and costs 350 rubles. The second type of gift contains 20 chocolate and 5 caramel candies and costs 500 rubles. Eugene wants to buy an equal number of caramel and chocolate candies, and these candies are only sold in the above-mentioned gift sets. What is the smallest non-zero amount of money he will have to spend?
Answer: 3750 rubles.

Solution. Consider integer values $m$ and $n$ - the quantities of purchased gift sets of candies of the 1st and 2nd types, respectively. These quantities must satisfy the conditions of the problem: the total number of caramel and chocolate candies in them is the same, and this number is the smallest possible under the conditions. Then, satisfying the conditions of the problem, we form an equation for the unknown quantities: $3 m + 20 n = 15 m + 5 n$

solving which as an equation in integers, we get the ratio $5 n = 4 m$,

from which we obtain a non-zero solution $n = 4 k, m = 5 k, k \in \mathbb{N}$. The smallest solution will be for $k=1$. That is, $m=5, n=4$. Then, the smallest cost of such a quantity of gifts will be $350 m + 500 n = 350 \cdot 5 + 500 \cdot 4 = 3750$ rubles.

Remarks on evaluation. Answer without justification - 0 points. Guessing the values of the quantities of gift sets without justifying the minimality - 2 points. Equation for the quantities of gift sets and candies is formed and the solutions are analyzed - 3 points.