5. (7 points) Three schoolgirls entered a store. Anya bought 2 pens, 7 pencils, and 1 notebook, Varya - 5 pens, 6 pencils, and 5 notebooks, Sasha - 8 pens, 4 pencils, and 9 notebooks. They all paid equally, but one of them used a discount when paying. Who? (Explain your answer).
Answer: Varya.

Solution. Let's denote the cost of a pen, a pencil, and a notebook by the symbols $\mathrm{P}, \mathrm{K}$, and L, respectively. We also denote the total cost of purchases (excluding the discount) by A, B, and C for Anya, Varya, and Sasha, respectively. According to the problem,

$$
A=2 \mathrm{P}+7 \mathrm{~K}+1 \mathrm{~L} ; B=5 \mathrm{P}+6 \mathrm{~K}+5 \mathrm{~L} ; C=8 \mathrm{P}+4 \mathrm{~K}+9 \mathrm{~L}
$$

Let's add the expressions for $A$ and $C$ and compare the result with $2 B$:

$$
A+C=10 \mathrm{P}+11 \mathrm{~K}+10 \mathrm{~L}<10 \mathrm{P}+12 \mathrm{~K}+10 \mathrm{~L}=2 B
$$

We obtained that $A+C<2 B$, which means that the cost of Varya's purchases is greater than the cost of at least one of the other girls' purchases, and therefore only Varya could have used the discount.

Criteria. Any correct solution: 7 points.

Only the correct answer is provided: 1 point.

Only the specific case of the prices of pens, pencils, and notebooks is considered: 0 points.