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Difference between revisions of "2022 AIME I Problems/Problem 7"

m (→‎Solution)
(The original solution's claim fraction >= 1/(7*8*9) is fairly clear, so there is no need to change "It is clear that ...".)
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==Solution==
 
==Solution==
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To minimize a positive fraction, we minimize its numerator and maximize its denominator. We must have that <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \geq \frac{1}{7\cdot8\cdot9}.</math>
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To minimize a positive fraction, we minimize its numerator and maximize its denominator. It is clear that <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \geq \frac{1}{7\cdot8\cdot9}.</math>
  
 
If we minimize the numerator, then <math>a \cdot b \cdot c - d \cdot e \cdot f = 1.</math> Note that <math>a \cdot b \cdot c \cdot d \cdot e \cdot f = (a \cdot b \cdot c) \cdot (a \cdot b \cdot c - 1) \geq 6! = 720,</math> so <math>a \cdot b \cdot c \geq 28.</math> It follows that <math>a \cdot b \cdot c</math> and <math>d \cdot e \cdot f</math> are consecutive composites with prime factors no other than <math>2,3,5,</math> and <math>7.</math> The smallest values for <math>a \cdot b \cdot c</math> and <math>d \cdot e \cdot f</math> are <math>36</math> and <math>35,</math> respectively. So, we have <math>\{a,b,c\} = \{2,3,6\}, \{d,e,f\} = \{1,5,7\},</math> and <math>\{g,h,i\} = \{4,8,9\},</math> from which <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} = \frac{1}{288}.</math>
 
If we minimize the numerator, then <math>a \cdot b \cdot c - d \cdot e \cdot f = 1.</math> Note that <math>a \cdot b \cdot c \cdot d \cdot e \cdot f = (a \cdot b \cdot c) \cdot (a \cdot b \cdot c - 1) \geq 6! = 720,</math> so <math>a \cdot b \cdot c \geq 28.</math> It follows that <math>a \cdot b \cdot c</math> and <math>d \cdot e \cdot f</math> are consecutive composites with prime factors no other than <math>2,3,5,</math> and <math>7.</math> The smallest values for <math>a \cdot b \cdot c</math> and <math>d \cdot e \cdot f</math> are <math>36</math> and <math>35,</math> respectively. So, we have <math>\{a,b,c\} = \{2,3,6\}, \{d,e,f\} = \{1,5,7\},</math> and <math>\{g,h,i\} = \{4,8,9\},</math> from which <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} = \frac{1}{288}.</math>
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Together, we conclude that the minimum possible positive value of <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}</math> is <math>\frac{1}{288}.</math> Therefore, the answer is <math>1+288=\boxed{289}.</math>
 
Together, we conclude that the minimum possible positive value of <math>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}</math> is <math>\frac{1}{288}.</math> Therefore, the answer is <math>1+288=\boxed{289}.</math>
  
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~MRENTHUSIASM ~jgplay ~PureSwag
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~MRENTHUSIASM ~jgplay
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=I|num-b=6|num-a=8}}
 
{{AIME box|year=2022|n=I|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:28, 7 March 2022

Problem

Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\] can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

To minimize a positive fraction, we minimize its numerator and maximize its denominator. It is clear that $\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \geq \frac{1}{7\cdot8\cdot9}.$

If we minimize the numerator, then $a \cdot b \cdot c - d \cdot e \cdot f = 1.$ Note that $a \cdot b \cdot c \cdot d \cdot e \cdot f = (a \cdot b \cdot c) \cdot (a \cdot b \cdot c - 1) \geq 6! = 720,$ so $a \cdot b \cdot c \geq 28.$ It follows that $a \cdot b \cdot c$ and $d \cdot e \cdot f$ are consecutive composites with prime factors no other than $2,3,5,$ and $7.$ The smallest values for $a \cdot b \cdot c$ and $d \cdot e \cdot f$ are $36$ and $35,$ respectively. So, we have $\{a,b,c\} = \{2,3,6\}, \{d,e,f\} = \{1,5,7\},$ and $\{g,h,i\} = \{4,8,9\},$ from which $\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} = \frac{1}{288}.$

If we do not minimize the numerator, then $a \cdot b \cdot c - d \cdot e \cdot f > 1.$ Note that $\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \geq \frac{2}{7\cdot8\cdot9} > \frac{1}{288}.$

Together, we conclude that the minimum possible positive value of $\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}$ is $\frac{1}{288}.$ Therefore, the answer is $1+288=\boxed{289}.$

~MRENTHUSIASM ~jgplay

See Also

2022 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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