Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem22

Video Transcription

Question number 22. 100 runners finished the race, each with a different time. When asked in what place they finished, the runners responded with a number between 1 and 100.  The sum of all these numbers was 4000. The smallest possible number of incorrect responses given by the runners equals to which of the following numbers?  So we have 100 numbers here, 100 integers from 1 to 100 and we can sum them.  So 1 plus 2 plus 3 plus all the way up to 100 can be computed by using the formula for the sum of the first n consecutive integers  and that formula is n times n plus 1 divided by 2 where here we're summing the first 100 integers so when n is equal to 100  and that gives us 100 times 101 divided by 2 which is 50 times 101 and that is 5050. And what we know is that the runners answers add to 4000 not the correct answer of 5050.  So some of them are wrong. And what we then need to do is in the question find the smallest possible number of incorrect responses.  So there is a difference between these and what we do is allocate the difference which is 5050 minus 4000 so 1050 between as few runners as possible.  So those claiming to have finished towards the end. So the last finishers. And then what we do is we compute.  So if we take 100 plus 99 plus 98 plus 97 the runners here that have come in or have claimed to finish towards the end  plus basically some value here of m until we are looking at a number that's strictly greater than 10,050.  So what we have to do is find the smallest value here of m counting down consecutively from 100.  And so it turns out that if m is equal to 90 we get the answer 10,045 but with the next value m equal to 89 what we get is 1134.  And 1134 here is bigger than 1050 so this is the answer we are looking for.  We obtain that when m is equal to 89 and that is a sum of 100 plus 99 plus 98 all the way up to m is equal to 89. So with m equal to 89 what we have here is exactly 12 terms.  And so it takes 12 of the highest times to sum to a number that exceeds the error in the runners claimed finish times. And that has to be the number of runners that made a mistake.  So we have our answer and that is 12 runners or d.

Video Summary

The problem involves 100 runners who finished a race, each providing their placement number totaling 4000 instead of the correct sum of 5050. To find the minimum number of incorrect responses, we calculate the difference of 1050 (5050 - 4000) and distribute it among the fewest high-numbered responses. By summing from 100 downwards, it is determined that 12 incorrect responses suffice, with 100 down to 89, which exceeds the required difference. Thus, the smallest number of incorrect responses is 12.

Keywords

race

runners

incorrect responses

placement numbers

sum discrepancy