1. In 100 containers of three types with capacities of $1 \pi$, 10 liters, and 50 liters, 500 liters of oil have been distributed. How many containers of each type were used, if the amount of oil in each container corresponds to its capacity?

(12 points)
Solution. Let $n, m, k$ be the number of containers with capacities of 1 liter, 10 liters, and 50 liters, respectively. Then $\left\{\begin{array}{c}n+10 m+50 k=500, \\ n+m+k=100 .\end{array}\right.$ Since $n=500-10 m-50 k=10(50-m-5 k)$, $n$ is divisible by 10, i.e., $n=10 l, l \in N, n \leq 100, l \leq 10 .\left\{\begin{array}{c}l+m+5 k=50, \\ 10 l+m+k=100,\end{array} \quad\left\{\begin{array}{c}l+m+5 k=50, \\ 9 l-4 k=50,\end{array} \Rightarrow l=5+(5+4 k) / 9\right.\right.$ , $(5+4 k) / 9 \leq 5,(5+4 k) / 9 \in N$. From this, we have $k \leq 10,(5+4 k) / 9 \in N, \Rightarrow$ $k=1, k=10 . \quad$ Then 1) $k=1, l=6, n=60, m=39 ; 2) \quad k=10, l=10, n=100, m<0, \quad$ there are no non-negative integer solutions.

Answer: 60 of 1 liter, 39 of 10 liters, 1 of 50 liters.