Difference between revisions of "2003 AIME II Problems/Problem 12"
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== 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>. |
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| + | Candidate <math>i</math> got <math>\frac{v_i}s</math> of the votes, hence the percentage of votes she 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>. | ||
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| + | 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>. | ||
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| + | 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>. | ||
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| + | 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>. | ||
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| + | 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>. | ||
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| + | 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. | ||
== See also == | == See also == | ||
{{AIME box|year=2003|n=II|num-b=11|num-a=13}} | {{AIME box|year=2003|n=II|num-b=11|num-a=13}} | ||
| + | v_1+\cdots+v_{27} | ||
Revision as of 22:58, 28 January 2009
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 she received is 
. The condition in the problem statement says that 
.
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.
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 | ||
v_1+\cdots+v_{27}
