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# 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 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 a
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. What is the smallest
possible number of members of the committee? So let's define some
variables here and maybe we can get our heads around this. So let's just say that m is
equal to the number of members of the committee. 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 one to 27, any integer. This sentence right here,
let me write in blue since I underlined it in blue. Let me write it here. Candidate c sub i is
equal to the number of votes for the ith 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-- OK. So 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 decimal. If you want it as a
percentage, you're going to want to
multiply it by 100. So times 100. So that's this part. The exact percentage of
votes the candidate got, so that's this right here,
was smaller by at least one than the number of votes
for that candidate. So smaller by at least one. So the number of votes of
the candidate is c sub i, and it's not less than or
equal to the number of votes they got. It's less than or equal to 1
less than the number of votes they got, at least 1 less. So we're going to
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 1,
which is really what that sentence is saying. And frankly, that's the hardest
part about this problem, is understanding what that
sentence is even saying. Now what is the
smallest possible number of members of the committee? Now think about it. 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
the minimum number of votes for each candidate,
per candidate. So can we have 0
votes for a candidate? Well, they tell us here. They say 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 per candidate? It seems fair that they
each got at least one vote. But if we look over
here, their percentage having to be 1 less than the
number of votes they have, and that's what we
wrote over here, that makes 1 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 is ringing. My apologies. So where I left off,
we were thinking, can we have one
vote for a candidate or can there be a candidate
that only has one vote? 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 1, this
right here would be 0. And this, on the left hand
side, if it's at least 1-- we know that we have a
positive number of members. And we know that this is
going to be at least 1. 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 0. So even the guy or the gal
who got the fewest votes has to get more than one vote. So this also can't be the case. This expression right
over here can never be less than or equal to
0, which would be the case if the minimum number
of votes was one. So let's think
about more of them. Let's see if we can
ever have two votes for a candidate, a
minimum of two votes. And to do that, what I want to
do-- if we put 2 in here-- what I want to do is I want to
see what repercussions that has for our members. Does it put any threshold
for our members? And to do that,
let me solve for m. It actually simplifies
a little bit now that we've put the constraint
that our minimum number of votes can never
be equal to 1. 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 100 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 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 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 100ci, 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 ith candidate
got over ci minus 1. So let's see. Can a candidate get 2 votes? So if a candidate got 2
votes-- if the candidate with the fewest votes
got 2, what repercussions 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
greater than or equal to 200. So maybe our answer's 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 does 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. Yep. It did come down. So it looks like as we increase
our minimum number of votes per candidate, we're
getting this threshold for m to come down. So let's try higher numbers. If our minimum vote per
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 133 1/3. And obviously, this has to
be an integer right here. So the smallest possible m in
this situation would be 134. Well, it seems like we're
making progress here. Let's try 5. If we have a minimum
number of votes of 5, then m has to be greater than
or equal to 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 about the minimum-- if this is the minimum
votes per candidate, what is the minimum
number of votes, not even thinking about
this threshold here. Well, it's going to be
this number times 27. In this case, if you had
two votes for candidate you would have to have 54. That's 27 times 2. 27 times 3 is 81. 27 times 4 is what. That's 80, that's 108. 27 times 5 is 135. So 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 135 votes. You can't get down to 125 votes. If you have 4 votes
per candidate-- in order to have 4 times
27, you only need 108. So if you bump that
108 to 134-- Well, there's two ways you
could get 134 votes. You can give 26 people
5 votes, and then give the last person
4 votes, and that would meet the
constraint over here. It gets you 1 less than 135. Or you could give
26 people 4 votes and give the last
person-- I think it would come out
to be 30 votes. And either way, this would work. So our answer is if
we have 134 votes-- or I guess another way to say
it is our answer to the question is the minimum number
of committee members we need is 134. Hopefully, you found
that interesting.