8.5. For breakfast, mom daily gives Seryozha either 1 sandwich and 3 candies, or 2 sandwiches and 4 candies, or 3 sandwiches and 5 candies. After several days, it turned out that Seryozha had eaten exactly 100 sandwiches. Could he have eaten exactly 166 candies during the same time? Justify your answer.
# Solution:

Method 1. Suppose this happened. Note that each day Petya eats 2 more candies than sandwiches. Therefore, the eating process took exactly $\frac{166-100}{2}=33$ days. But in 33 days, Petya could not have eaten more than $3 \cdot 33=99$ sandwiches. Contradiction.

Method 2. Let $x$ be the number of days Seryozha eats 1 sandwich and 3 candies, $y$ be the number of days he eats 2 sandwiches and 4 candies, and $z$ be the number of days he eats 3 sandwiches and 5 candies. Then we have $x+2 y+3 z=100$, and the number of candies eaten is $3 x+4 y+5 z$. We solve the system $\left\{\begin{aligned} x+2 y+3 z & =100 \\ 3 x+4 y+5 z & =166\end{aligned}\right.$ in non-negative integers. Subtract the first equation from the second. After dividing by 2, we get $x+y+z=33$, from which $z=33-x-y$. Substitute the found value of $z$ into the first equation: $x+2 y+3(33-x-y)=100$, which simplifies to $-2 x-y=1$. Contradiction, as the left side is negative while the right side is positive. Therefore, the system has no solutions in non-negative integers, and Seryozha could not have eaten exactly 166 candies.

Note: All solutions to the system are triples of the form $(t, -2t-1, t+34)$. The contradiction lies in the impossibility of simultaneously satisfying the inequalities $t \geqslant 0$ and $-2t-1 \geqslant 0$.

Answer: He could not.

Recommendations for checking:

| present in the work | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| It is noted that the difference between the number of candies and sandwiches eaten increases by exactly 2 each day | 3 points |
| The problem is correctly reduced to the analysis of a system of linear equations, but this analysis is not carried out | 2 points |
| Answer without justification or illustration of answers with specific examples (in any number) | 0 points |