Difference between revisions of "2003 AIME II Problems/Problem 12"
I_like_pie (talk | contribs) |
m (→Solution: ; clean up) |
||
(14 intermediate revisions by 9 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
+ | The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of the committee? | ||
== Solution == | == Solution == | ||
− | {{ | + | Let <math>v_i</math> be the number of votes candidate <math>i</math> received, and let <math>s=v_1+\cdots+v_{27}</math> be the total number of votes cast. Our goal is to determine the smallest possible <math>s</math>. |
+ | |||
+ | Candidate <math>i</math> got <math>\frac{v_i}s</math> of the votes, hence the percentage of votes they received is <math>\frac{100v_i}s</math>. The condition in the problem statement says that <math>\forall i: \frac{100v_i}s + 1 \leq v_i</math>. (<math>\forall</math> means "for all", so this means "For all <math>i</math>, <math>\frac{100v_i}s + 1 \leq v_i</math> is true") | ||
+ | |||
+ | Obviously, if some <math>v_i</math> would be <math>0</math> or <math>1</math>, the condition would be false. Thus <math>\forall i: v_i\geq 2</math>. We can then rewrite the above inequality as <math>\forall i: s\geq\frac{100v_i}{v_i-1}</math>. | ||
+ | |||
+ | If for some <math>i</math> we have <math>v_i=2</math>, then from the inequality we just derived we would have <math>s\geq 200</math>. | ||
+ | |||
+ | If for some <math>i</math> we have <math>v_i=3</math>, then <math>s\geq 150</math>. | ||
+ | |||
+ | And if for some <math>i</math> we have <math>v_i=4</math>, then <math>s\geq \frac{400}3 = 133\frac13</math>, and hence <math>s\geq 134</math>. | ||
+ | |||
+ | Is it possible to have <math>s<134</math>? We just proved that to have such <math>s</math>, all <math>v_i</math> have to be at least <math>5</math>. But then <math>s=v_1+\cdots+v_{27}\geq 27\cdot 5 = 135</math>, which is a contradiction. Hence the smallest possible <math>s</math> is at least <math>134</math>. | ||
+ | |||
+ | Now consider a situation where <math>26</math> candidates got <math>5</math> votes each, and one candidate got <math>4</math> votes. In this situation, the total number of votes is exactly <math>134</math>, and for each candidate the above inequality is satisfied. Hence the minimum number of committee members is <math>s=\boxed{134}</math>. | ||
+ | |||
+ | Note: Each of the <math>26</math> candidates received <math>\simeq 3.63\%</math> votes, and the last candidate received <math>\simeq 2.985\%</math> votes. | ||
+ | |||
+ | == Solution 2 == | ||
+ | Let there be <math>N</math> members of the committee. | ||
+ | Suppose candidate <math>n</math> gets <math>a_n</math> votes. | ||
+ | Then <math>a_n</math> as a percentage out of <math>N</math> is | ||
+ | <math>100\frac{a_n}{N}</math>. Setting up the inequality | ||
+ | <math>a_n \geq 1 + 100\frac{a_n}{N}</math> and simplifying, | ||
+ | <math>a_n \geq \lceil(\frac{N}{N - 100})\rceil</math> (the ceiling function is there because <math>a_n</math> is an integer. | ||
+ | Note that if we set all <math>a_i</math> equal to <math>\lceil(\frac{N}{100 - N})\rceil</math> we have <math>N \geq 27\lceil(\frac{N}{100 - N})\rceil</math>. Clearly <math>N = 134</math> is the least such number that satisfies this inequality. Now we must show that we can find suitable <math>a_i</math>. We can let 26 of them equal to <math>5</math> and one of them equal to <math>4</math>. Therefore, <math>N = \boxed{134}</math> is the answer. | ||
+ | - whatRthose | ||
+ | |||
+ | == Solution 3 == | ||
+ | Let <math>n</math> be the total number of people in the committee, and <math>a_i</math> be the number of votes candidate <math>i</math> gets where <math>1 \leq i \leq 27</math>. The problem tells us that <cmath>\frac{100a_i}{n} \leq a_i - 1 \implies 100a_i \leq na_i - n \implies a_i \geq \frac{n}{n-100}.</cmath>Therefore, <cmath>\sum^{27}_{i=1} a_i = n \geq \sum^{27}_{i=1} \frac{n}{n-100} = \frac{27n}{n-100},</cmath>and so <math>n(n-127) \geq 0 \implies n \geq 127</math>. Trying <math>n = 127</math>, we get that <cmath>a_i \geq \frac{127}{27} \approx 4.7 \implies a_i \geq 5 \implies \sum^{27}_{a_i} a_i \geq 5 \cdot 27 = 135 \geq 127,</cmath>a contradiction. Bashing out a few more, we find that <math>\boxed{n = 134}</math> works perfectly fine. | ||
+ | |||
+ | == Video Solution by Sal Khan == | ||
+ | https://www.youtube.com/watch?v=KD46pC_KFWk&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=10 | ||
+ | - AMBRIGGS | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=2003|n=II|num-b=11|num-a=13}} | |
− | |||
− | |||
− | + | [[Category: Intermediate Algebra Problems]] | |
+ | {{MAA Notice}} |
Latest revision as of 13:00, 11 December 2022
Problem
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of the committee?
Solution
Let be the number of votes candidate
received, and let
be the total number of votes cast. Our goal is to determine the smallest possible
.
Candidate got
of the votes, hence the percentage of votes they received is
. The condition in the problem statement says that
. (
means "for all", so this means "For all
,
is true")
Obviously, if some would be
or
, the condition would be false. Thus
. We can then rewrite the above inequality as
.
If for some we have
, then from the inequality we just derived we would have
.
If for some we have
, then
.
And if for some we have
, then
, and hence
.
Is it possible to have ? We just proved that to have such
, all
have to be at least
. But then
, which is a contradiction. Hence the smallest possible
is at least
.
Now consider a situation where candidates got
votes each, and one candidate got
votes. In this situation, the total number of votes is exactly
, and for each candidate the above inequality is satisfied. Hence the minimum number of committee members is
.
Note: Each of the candidates received
votes, and the last candidate received
votes.
Solution 2
Let there be members of the committee.
Suppose candidate
gets
votes.
Then
as a percentage out of
is
. Setting up the inequality
and simplifying,
(the ceiling function is there because
is an integer.
Note that if we set all
equal to
we have
. Clearly
is the least such number that satisfies this inequality. Now we must show that we can find suitable
. We can let 26 of them equal to
and one of them equal to
. Therefore,
is the answer.
- whatRthose
Solution 3
Let be the total number of people in the committee, and
be the number of votes candidate
gets where
. The problem tells us that
Therefore,
and so
. Trying
, we get that
a contradiction. Bashing out a few more, we find that
works perfectly fine.
Video Solution by Sal Khan
https://www.youtube.com/watch?v=KD46pC_KFWk&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=10 - AMBRIGGS
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.