Ответ. 24.

Решение. Upper bound. We will show that $k \le 24$, i.e., that any expert could be incompetent. If this expert believes that all athletes will receive one medal each, we can refute them with the result $(25, 0, 0, ..., 0)$. Otherwise, the expert believes that several (at least one) athletes will receive 0 medals. Then we can distribute all medals among these athletes so that each of them receives at least one medal. In this case, the expert won't guess any medal count correctly.

Example. Let one expert's prediction be $(1, 1, 1, ..., 1)$, and the predictions of others be $(1, 0, ..., 0, 24)$, $(0, 1, ..., 0, 24)$, ..., $(0, 0, ..., 1, 24)$ (with 24 in the last position and one more 1).

If the first expert is incompetent, then the actual result must contain at least three zeros. Otherwise, at least 23 positions would have at least 2 medals, making the total medal count at least $23 \cdot 2 > 30$ - a contradiction. But then all other experts must be competent.

Now suppose two experts other than the first are incompetent. Then in two positions their predictions are 0 and 1 medals, meaning in the actual result these positions have at least 2 medals. Moreover, in 22 other positions both experts predicted zeros, so in reality these positions have at least 1 medal. Thus the total medal count would be at least $2 \cdot 2 + 22 \cdot 1 > 25$ - a contradiction.