Difference between revisions of "2022 AIME I Problems"
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==Problem 7== | ==Problem 7== | ||
Let <math>a, b, c, d, e, f, g, h, i</math> be distinct integers from <math>1</math> to <math>9</math>. The minimum possible positive value of <cmath>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}</cmath>can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | Let <math>a, b, c, d, e, f, g, h, i</math> be distinct integers from <math>1</math> to <math>9</math>. The minimum possible positive value of <cmath>\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}</cmath>can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
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[[2022 AIME I Problems/Problem 7|Solution]] | [[2022 AIME I Problems/Problem 7|Solution]] | ||
Revision as of 18:26, 17 February 2022
| 2022 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
Problem 1
Quadratic polynomials 
and 
have leading coefficients 
and 
respectively. The graphs of both polynomials pass through the two points 
and 
Find 
Problem 2
Find the three-digit positive integer 
whose representation in base nine is 
where 
and 
are (not necessarily distinct) digits.
Problem 3
In isosceles trapezoid 
parallel bases 
and 
have lengths 
and 
respectively, and 
The angle bisectors of 
and 
meet at 
and the angle bisectors of 
and 
meet at 
Find 
Problem 4
Let 
and 
where 
Find the number of ordered pairs 
of positive integers not exceeding 
that satisfy the equation 
Problem 5
A straight river that is 
meters wide flows from west to east at a rate of 
meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of 
meters downstream from Sherry. Relative to the water, Melanie swims at 
meters per minute, and Sherry swims at 
meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find 
Problem 6
Find the number of ordered pairs of integers 
such that the sequence ![\[3,4,5,a,b,30,40,50\]](http://latex.artofproblemsolving.com/1/0/f/10f3618dca767ca85f66cb8fd4b671468c6e2317.png)
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Problem 7
Let 
be distinct integers from 
to 
. The minimum possible positive value of ![\[\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\]](http://latex.artofproblemsolving.com/d/d/f/ddfa7b426fa7e26bca77e14d125240631ed16b80.png)
can be written as 
where 
and 
are relatively prime positive integers. Find 
Problem 8
![\[\textbf{Please do not post this problem until the contest is released.}\]](http://latex.artofproblemsolving.com/6/8/b/68b95cde68b1140211f4700c09dba3f89cc56b5a.png)
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
See also
| 2022 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by 2021 AIME II |
Followed by 2022 AIME II | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
