For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0  \le  x_1 \le ...  \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards?

(Ilya Bogdanov)
1. **Initial Setup and Strategy**:
   - We have 10 variables \( x_1, x_2, \ldots, x_{10} \).
   - Each card has the product of 5 distinct variables.
   - Peter and Basil take turns picking cards, starting with Peter.
   - After all cards are taken, Basil assigns values to the variables such that \( 0 \le x_1 \le x_2 \le \cdots \le x_{10} \).

2. **Basil's Winning Strategy**:
   - Basil can ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards by carefully choosing the values of the variables after all cards are taken.

3. **Key Observations**:
   - If Peter does not take the card with the product \( x_6 x_7 x_8 x_9 x_{10} \), Basil can choose this card and set \( x_1 = x_2 = x_3 = x_4 = x_5 = 0 \) and \( x_6 = x_7 = x_8 = x_9 = x_{10} = \pi^{1 + \frac{2e}{\pi}} \). This would make the product on Basil's card very large while making the products on Peter's cards zero.
   - If Peter takes the card with \( x_6 x_7 x_8 x_9 x_{10} \), Basil can choose the card with \( x_5 x_7 x_8 x_9 x_{10} \) and set \( x_1 = x_2 = x_3 = x_4 = 0 \), \( x_5 = x_6 = x_7 = e \), and \( x_8 = x_9 = x_{10} = \pi^{2 \cdot \pi^\pi} \).

4. **Detailed Calculation**:
   - Suppose Peter takes \( x_6 x_7 x_8 x_9 x_{10} \). Basil then takes \( x_5 x_7 x_8 x_9 x_{10} \).
   - Basil sets \( x_1 = x_2 = x_3 = x_4 = 0 \), \( x_5 = x_6 = x_7 = e \), and \( x_8 = x_9 = x_{10} = \pi^{2 \cdot \pi^\pi} \).
   - Basil's product: \( x_5 x_7 x_8 x_9 x_{10} = e \cdot e \cdot (\pi^{2 \cdot \pi^\pi})^3 = e^2 \cdot \pi^{6 \cdot \pi^\pi} \).
   - Peter's product: \( x_6 x_7 x_8 x_9 x_{10} = e \cdot e \cdot (\pi^{2 \cdot \pi^\pi})^3 = e^2 \cdot \pi^{6 \cdot \pi^\pi} \).

5. **Ensuring Basil's Win**:
   - Basil can always choose cards such that the sum of the products on his cards is greater than the sum of the products on Peter's cards by setting the values of the variables appropriately.
   - For example, if Peter takes \( x_5 x_6 x_8 x_9 x_{10} \), Basil can take \( x_4 x_7 x_8 x_9 x_{10} \) and set \( x_1 = x_2 = x_3 = 0 \), \( x_4 = x_5 = x_6 = e \), and \( x_7 = x_8 = x_9 = x_{10} = \pi^{2 \cdot \pi^\pi} \).

6. **Conclusion**:
   - By carefully choosing the values of the variables after all cards are taken, Basil can ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards.

\[
\boxed{\text{Basil can ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards.}}
\]