Difference between revisions of "2022 AIME I Problems"
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{{AIME Problems|year=2022|n=I}} | {{AIME Problems|year=2022|n=I}} | ||
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==Problem 1== | ==Problem 1== | ||
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| + | Quadratic polynomials <math>P(x)</math> and <math>Q(x)</math> have leading coefficients <math>2</math> and <math>-2,</math> respectively. The graphs of both polynomials pass through the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> | ||
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[[2022 AIME I Problems/Problem 1|Solution]] | [[2022 AIME I Problems/Problem 1|Solution]] | ||
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==Problem 2== | ==Problem 2== | ||
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| + | Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits. | ||
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[[2022 AIME I Problems/Problem 2|Solution]] | [[2022 AIME I Problems/Problem 2|Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
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| + | In isosceles trapezoid <math>ABCD,</math> parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> have lengths <math>500</math> and <math>650,</math> respectively, and <math>AD=BC=333.</math> The angle bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P,</math> and the angle bisectors of <math>\angle B</math> and <math>\angle C</math> meet at <math>Q.</math> Find <math>PQ.</math> | ||
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[[2022 AIME I Problems/Problem 3|Solution]] | [[2022 AIME I Problems/Problem 3|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
<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 16:18, 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
![\[\textbf{Please do not post this problem until the contest is released.}\]](http://latex.artofproblemsolving.com/6/8/b/68b95cde68b1140211f4700c09dba3f89cc56b5a.png)
Problem 5
Problem 6
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. 
