Before the Christmas concert, the students offered for sale 60 products from their art classes. Each customer could determine the price themselves, and the entire proceeds went to charitable causes. At the beginning of the concert, the students calculated how many crowns they had earned on average for each sold product, and it came out to be an exact whole number. Since they had not yet sold all 60 products, they continued to offer them after the concert. Then, people bought seven more, for which they paid a total of 2505 Kč. This increased the average revenue per sold product to exactly 130 Kč. How many products were then left unsold?

(L. Šimünek)

Idea. Determine the relationship between the number of products sold before the concert and the amount earned from them. Keep in mind that all unknowns should be whole numbers.
Let the number of items the students sold before the concert be denoted by $v$, and the amount in crowns they received for them be denoted by $k$. According to the problem, the ratio $\frac{k}{v}$ is an integer and the following equation holds:

$$
\frac{k+2505}{v+7}=130
$$

From this equation, we express the unknown $k$ in terms of the unknown $v$:

$$
\begin{aligned}
k+2505 & =130 v+130 \cdot 7 \\
k & =130 v-1595 .
\end{aligned}
$$

We substitute the obtained expression into the fraction $\frac{k}{v}$ and then partially divide it:

$$
\frac{130 v-1595}{v}=130-\frac{1595}{v} .
$$

The unknown $v$ is a natural number, and since the above expression must be equal to an integer, $v$ must be a divisor of the number 1595. The number $1595=5 \cdot 11 \cdot 29$ has the following divisors:

$$
1,5,11,29,55,145,319,1595
$$

According to the problem, after selling $v$ items out of a total of 60, at least 7 remained, so $v \leqq 53$. If we substitute $v = 1, 5,$ or $11$ into the equation $k=130 v-1595$, we get a negative $k$, which contradicts the problem. For $v=29$, the revenue $k$ is positive, so the only acceptable value for $v$ is 29. In total, $29+7=36$ items were sold, and $60-36=24$ remained unsold.

Another idea. If we talk about the average price, try to simplify the situation by imagining that everyone mentioned paid exactly this price.

Another solution. For the purpose of solving, we can simplify the situation and imagine that each customer buying before the concert paid the same integer amount. This amount was equal to the average price calculated by the students before the concert.

After the concert, seven customers came and for each previous customer, they made up the difference to 130 Kč and paid $7 \cdot 130$ Kč for themselves. The total amount of the price adjustment thus corresponded to the amount $2505-7 \cdot 130=1595$ (Kč).

We need to factorize this amount into a product where one factor is the number of customers before the concert and the other factor is the number of crowns each of these customers was made up for by the later arrivals. We know from the problem that the first mentioned factor must be less than or equal to 53, and the second must be less than 130. The number 1595 can be factorized into two natural numbers in the following ways:

$$
1 \cdot 1595,5 \cdot 319,11 \cdot 145,29 \cdot 55
$$

The only factorization that meets the given conditions is $29 \cdot 55$. Therefore, a total of $29+7=36$ items were sold, and $60-36=24$ remained unsold.