Difference between revisions of "2022 AIME I Problems"

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==Problem 4==
 
==Problem 4==
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
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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==
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
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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==
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
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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)
Printable version | AoPS Contest CollectionsPDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Problem 1

Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$

Solution

Problem 2

Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits.

Solution

Problem 3

In isosceles trapezoid $ABCD,$ parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650,$ respectively, and $AD=BC=333.$ The angle bisectors of $\angle A$ and $\angle D$ meet at $P,$ and the angle bisectors of $\angle B$ and $\angle C$ meet at $Q.$ Find $PQ.$

Solution

Problem 4

Let $w = \dfrac{\sqrt{3} + i}{2}$ and $z = \dfrac{-1 + i\sqrt{3}}{2},$ where $i = \sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \cdot w^r = z^s.$

Solution

Problem 5

A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ 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 $D.$

Solution

Problem 6

Find the number of ordered pairs of integers $(a,b)$ such that the sequence \[3,4,5,a,b,30,40,50\] is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

Solution

Problem 7

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

Problem 8

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

Problem 9

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Problem 10

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Problem 11

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Problem 12

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

Problem 13

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

Problem 14

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

Problem 15

\[\textbf{Please do not post this problem until the contest is released.}\] Solution

See also

2022 AIME I (ProblemsAnswer KeyResources)
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

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