Difference between revisions of "2001 AIME II Problems/Problem 2"
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== Problem == | == Problem == | ||
+ | Each of the <math>2001</math> students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between <math>80</math> percent and <math>85</math> percent of the school population, and the number who study French is between <math>30</math> percent and <math>40</math> percent. Let <math>m</math> be the smallest number of students who could study both languages, and let <math>M</math> be the largest number of students who could study both languages. Find <math>M-m</math>. | ||
== Solution == | == Solution == | ||
+ | Let <math>S</math> be the percent of people who study Spanish, <math>F</math> be the number of people who study French, and let <math>S \cap F</math> be the number of students who study both. Then <math>\left\lceil 80\% \cdot 2001 \right\rceil = 1601 \le S \le \left\lfloor 85\% \cdot 2001 \right\rfloor = 1700</math>, and <math>\left\lceil 30\% \cdot 2001 \right\rceil = 601 \le F \le \left\lfloor 40\% \cdot 2001 \right\rfloor = 800</math>. By the [[Principle of Inclusion-Exclusion]], | ||
+ | |||
+ | <cmath>S+F- S \cap F = S \cup F = 2001 </cmath> | ||
+ | |||
+ | For <math>m = S \cap F</math> to be smallest, <math>S</math> and <math>F</math> must be minimized. | ||
+ | |||
+ | <cmath>1601 + 601 - m = 2001 \Longrightarrow m = 201</cmath> | ||
+ | |||
+ | For <math>M = S \cap F</math> to be largest, <math>S</math> and <math>F</math> must be maximized. | ||
+ | |||
+ | <cmath>1700 + 800 - M = 2001 \Longrightarrow M = 499</cmath> | ||
+ | |||
+ | Therefore, the answer is <math>M - m = 499 - 201 = \boxed{298}</math>. | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=2001|n=II|num-b=1|num-a=3}} | |
+ | |||
+ | [[Category:Intermediate Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 00:54, 10 January 2024
Problem
Each of the students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between
percent and
percent of the school population, and the number who study French is between
percent and
percent. Let
be the smallest number of students who could study both languages, and let
be the largest number of students who could study both languages. Find
.
Solution
Let be the percent of people who study Spanish,
be the number of people who study French, and let
be the number of students who study both. Then
, and
. By the Principle of Inclusion-Exclusion,
For to be smallest,
and
must be minimized.
For to be largest,
and
must be maximized.
Therefore, the answer is .
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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