Problem 1619

1. Initial Setup and Problem Understanding:
- We have a 10 × 10 table with positive numbers.
- Frogs sit on five squares, covering the numbers in these squares.
- Kostya calculates the sum of all visible numbers and gets 10.
- Each frog jumps to an adjacent square, and the sum changes to $10^2$.
- This pattern continues, with each new sum being 10 times greater than the previous one.

2. Formulating the Problem:
- Let $S_0$ be the initial sum of visible numbers, which is 10.
- After the first jump, the sum becomes $S_1 = 10^2 = 100$.
- After the second jump, the sum becomes $S_2 = 10^3 = 1000$.
- This pattern continues up to $S_6 = 10^6$.

3. Analyzing the Sum Changes:
- The increase from $S_0$ to $S_1$ is $100 - 10 = 90$.
- Since 5 squares are revealed and 5 are covered, the net change in the sum is due to the difference in the values of these squares.
- Let $a_1, a_2, a_3, a_4, a_5$ be the values of the squares initially covered by the frogs.
- Let $b_1, b_2, b_3, b_4, b_5$ be the values of the new squares covered by the frogs after the first jump.
- The change in sum can be expressed as:
$$S_1 - S_0 = (b_1 + b_2 + b_3 + b_4 + b_5) - (a_1 + a_2 + a_3 + a_4 + a_5)$$
Given $S_1 = 100$ and $S_0 = 10$, we have:
$$100 - 10 = (b_1 + b_2 + b_3 + b_4 + b_5) - (a_1 + a_2 + a_3 + a_4 + a_5) = 90$$

4. Bounding the Values:
- Since the sum increases by a factor of 10 each time, the values of the revealed squares must be significantly larger than the values of the covered squares.
- For the increase from $10^k$ to $10^{k+1}$, the increase is $10^{k+1} - 10^k = 9 \times 10^k$.
- At least one of the revealed squares must have a value greater than $\frac{9 \times 10^k}{5} = 1.8 \times 10^k$.

5. Proving the Maximum Sum:
- For the sum to reach $10^7$, the increase from $10^6$ to $10^7$ would be $9 \times 10^6$.
- This would require at least one square to have a value greater than $1.8 \times 10^6$.
- However, since there are only 5 frogs and each jump reveals 5 new squares, the values of these squares must be unique and within specific ranges.
- The ranges for the values of the revealed squares are:
$$(1.5 \times 10^2, 10 \times 10^2), (1.5 \times 10^3, 10 \times 10^3), \ldots, (1.5 \times 10^6, 10 \times 10^6)$$
- These ranges are non-intersecting, meaning each value must be unique.
- Since there are only 5 frogs, at least one square must be visible at all times, which is impossible given the constraints.

6. Conclusion:
- Therefore, Kostya cannot achieve a sum of $10^7$.
- The maximum sum Kostya can obtain is $10^6$.

The final answer is $\boxed{10^6}$