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Current time:0:00Total duration:9:36

2003 AIME II problem 12

Video transcript

the members of a distinguished committee were choosing a president and each member gave one vote to one of the 27 candidates so each member gave one vote to one of the 27 candidates so it looks like they all got at least one vote for each candidate the exact percentage of votes the candidate got was smaller by at least one than the number of votes for that candidate let me underline this again and read it again it's kind of confusing statement for each candidate the exact percentage of votes the candidate got was smaller by at least one than the number of votes for that candidate was smaller by at least one than the number of votes for that candidate what is the smallest possible number of members of the committee so let's let's define some variables here and maybe we can get our get our heads around this so let's just say that M is equal to the number of members of the committee members that's equal to the number of members and let's just think about each candidate let's say that we have 27 candidates but let's just pick some arbitrary candidate I so that I could be any number from 1 to 27 any integer so candidate I this sentence right here let me write it in blue since I underlined it in blue let's let me write here candidate C sub I is equal the number of votes for the 8th for the 8th candidate for the ithe candidate now let's see if we can do something with what I underlined in blue the exact percentage of votes the candidate got was smaller okay so let's the exact percentage of votes the candidate got would be the votes he got or she got divided by the total number of members now this will give you a number this will give you a decimal if you wanted as a percentage if you wanted as a percentage you're going to want to multiply it by a hundred so times 100 so that's what this part the exact percentage of votes the candidate got so that's this right here was smaller by at least one then the number of votes that for that candidate so smaller than SMO by at least one so the number of votes of the candidate is C sub I C sub I and it's smaller and it's not less than or equal to the number of votes they got it's less it's smaller than its less than or equal to 1 less than the number of votes they got at least one less was smaller by at least one so we're gonna put a minus 1 here so this is telling us that the percentage of votes that any candidate got is less than the number of votes they got minus one which is really what that sense to saying and frankly that's the hardest part about this problem is understanding what that sense is even saying now what is the smallest possible number of members of the committee now the thing about we want to minimize the number of members so we really want to minimize the number of votes that each candidate gets so let's just think about let's just think about the minimum number of votes minimum the minimum votes for each candidate per per candidate per candidate so can we have 0 votes per candidate well they tell us here they say and each member gave one vote to one of the 27 candidates so everyone got at least one vote so no we cannot have 0 votes can we have one vote one vote per candidate it seems fair that they each got at least one vote but if we look over here this their percentage having to be less than 1 less than the number of votes they have and that's what we wrote over here that makes one pretty difficult let's think about this a little bit let's think about it if the minimum number of votes is 1 that means for that candidate who got the minimum number of votes the right-hand side of this equation right here this right-hand side of the equation would be sorry my cell phone was ringing my apologies so where I left off we were thinking can we have one vote for Canada or can there be a candidate that only has one vote is can the minimum votes per candidate be one and we were looking at the right-hand side of this equation we set up from the word problem and if it was one this right here would be zero and this on the left-hand side if it's at least one we know that we have a positive number of members and we know that this is going to be at least one so this is going to be some positive fraction that we're going to multiply by 100 so there's no way that that can be less than or equal to zero so you can you can't have even the guy who got or the gal who got the fewest votes has to get more than one vote so this also this also can't be the case this expression right over here can never can never be less than or equal to zero it should be the case if the minimum number of votes was one so let's think about more of them let's think about let's see if we can ever have two votes per candidate a minimum of two votes and to do that what I want to do if we put two in here what I want to do is I want to see what what repercussions that has for our members does it put any threshold for members and do that let me solve for M it actually simplifies a little bit now that we put the constraint that the our minimum number of votes can never be it can never be equal to one so what I want to do is let's take the inverse of both sides of this equation let me do this in a new color let me do it in pink let's take the inverse of both sides of this equation the left-hand side over here we essentially have a hundred CI over m so that will be M over 100 C sub I and since we're inverting both sides of the equation this less than or equal is going to become a greater than or equal to and then we're going to have 1 over 1 over C sub I minus 1 so I just inverted both sides you swap the inequality obviously if you had 1 is less than 2 now if you invert them 1 is greater than or equal to or one is greater than 1/2 so that's the logic behind swapping the inequality and now if I want to solve for M I can multiply both sides by a hundred C I and that's a positive value so it won't do anything to the inequality so M is going to be greater than or equal to 100 C I the number of votes the ithe candidate got over C I minus 1 so let's see can a candidate get two votes so if a candidate got two votes it's the lift if the candidate the fewest votes got to what would what repercussions would that over here would that have based on what we just solved for well in that situation M would have to be greater than or equal to 100 times 2 over 2 minus 1 so M would have to be greater than 200 seems reasonable maybe our answer is 200 or Ida greater than or equal to 200 so maybe our answer is 200 but let's try it with some larger minimum number of votes maybe that can bring down our M a little bit so what if we had 3 what is that what is this inequality tell us then M would have to be greater than or equal to 300 divided by 3 minus 1 300 divided by 2 yup it did come down so it looks like as we increase as we increase our minimum number of votes per candidate we're getting this threshold for M to calm down so let's try higher numbers if our minimum vote for candidate is 4 then we have M is greater than or equal to 400 over 3 so M would have to be greater than or equal to 130 130 3 and 1/3 and obviously this has to be an integer right here so the smallest possible M in this situation would be 134 well where it seems like we're making progress here let's try 5 let's try 5 we have if we have a minimum number of votes of 5 then M has to be greater than or equal to 100 500 divided by 4 is 125 now this looks tempting but remember this is the minimum number of votes per candidate and we have 27 candidates so what does this tell us what does this tell us about the minimum if this is a minimum of votes for candidate what is the minimum number of votes not even thinking about this threshold here what's going to be this number times 27 in this case where if you had 2 votes per candidate you would have to have you would have to have 54 that's 27 times 2 27 times 3 is 81 27 times 4 is what that's 80 it's 108 108 27 times 5 is 135 so even though even though having at least 5 votes per candidate kind of loosens your constraint on M if you have 5 votes per candidate you're going to have a hundred and thirty-five votes you can't get down to 125 votes if you have 4 votes per candidate in order to have 4 times 20 seven you only need 108 so if you bump that hundred eight to three hundred and thirty four so essentially you could well there's two ways you could get 134 votes you can give 26 people 26 people five votes and then give the last person four votes and that would meet the constraint over here it gets you one less than 135 or you could give 26 people for votes and give the last person I think it would come out to be it would come out to be thirty votes and either way this would work so our answer is if we have a hundred and thirty-four votes or I guess another way to say our answer the question is the minimum number of committee members we need is 134 hopefully you found that interesting