Task B-4.5. In the race, 100 people participated, and no two people finished the race with the same time. At the end of the race, each participant was asked what place they finished in, and everyone answered with a number between 1 and 100.

The sum of all the answers is 4000. What is the smallest number of incorrect answers that the runners could have given? Justify your answer.
Solution. If all the runners had accurately answered their finishing positions, the sum of their answers would be

$$
1+2+3+\ldots+100=5050
$$

Thus, some people must have exaggerated their results, and the smallest number of incorrect answers is achieved when the runners at the last positions claim they were first.

If $n$ people at the end of the line lied that they were first, the sum of the answers is

$$
1+2+\ldots+(100-n)+1+1+\ldots+1=\frac{(100-n)(100-n+1)}{2}+n
$$

By direct verification, we can determine that for $n=11$ the sum is 4016, which is greater than 4000, and for $n=12$ the sum is 3928.

Therefore, to have the sum exactly 4000 with the smallest number of incorrect answers, 11 people at the end of the line need to say they were first, and since the difference to 4016 is 16, it is sufficient for the runner with the number 17 to say he was first.

Thus, a minimum of 12 runners gave incorrect answers.