Difference between revisions of "2022 AIME I Problems"
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==Problem 4== | ==Problem 4== | ||
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+ | Let <math>w = \dfrac{\sqrt{3} + i}{2}</math> and <math>z = \dfrac{-1 + i\sqrt{3}}{2},</math> where <math>i = \sqrt{-1}.</math> Find the number of ordered pairs <math>(r,s)</math> of positive integers not exceeding <math>100</math> that satisfy the equation <math>i \cdot w^r = z^s.</math> | ||
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[[2022 AIME I Problems/Problem 4|Solution]] | [[2022 AIME I Problems/Problem 4|Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
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+ | A straight river that is <math>264</math> meters wide flows from west to east at a rate of <math>14</math> meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of <math>D</math> meters downstream from Sherry. Relative to the water, Melanie swims at <math>80</math> meters per minute, and Sherry swims at <math>60</math> 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 <math>D.</math> | ||
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[[2022 AIME I Problems/Problem 5|Solution]] | [[2022 AIME I Problems/Problem 5|Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
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+ | Find the number of ordered pairs of integers <math>(a,b)</math> such that the sequence <cmath>3,4,5,a,b,30,40,50</cmath> is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. | ||
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[[2022 AIME I Problems/Problem 6|Solution]] | [[2022 AIME I Problems/Problem 6|Solution]] | ||
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==Problem 7== | ==Problem 7== | ||
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath> | <cmath>\textbf{Please do not post this problem until the contest is released.}</cmath> |
Revision as of 16:19, 17 February 2022
2022 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Problem 7
Problem 8
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.