Difference between revisions of "2022 AIME I Problems/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> | 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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+ | == Video Solution == | ||
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+ | https://youtu.be/MJ_M-xvwHLk?t=7 ~ ThePuzzlr | ||
==Solution 1 (Linear Polynomials)== | ==Solution 1 (Linear Polynomials)== |
Revision as of 16:16, 24 March 2022
Contents
[hide]Problem
Quadratic polynomials and have leading coefficients and respectively. The graphs of both polynomials pass through the two points and Find
Video Solution
https://youtu.be/MJ_M-xvwHLk?t=7 ~ ThePuzzlr
Solution 1 (Linear Polynomials)
Let Since the -terms of and cancel, we conclude that is a linear polynomial.
Note that so the slope of is
It follows that the equation of is for some constant and we wish to find
We substitute into this equation to get from which
~MRENTHUSIASM
Solution 2 (Quadratic Polynomials)
Let for some constants and
We are given that and we wish to find We need to cancel and Since we subtract from to get ~MRENTHUSIASM
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=sUfbEBCQ6RY
Video Solution by MRENTHUSIASM (English & Chinese)
https://www.youtube.com/watch?v=XcS5qcqsRyw&ab_channel=MRENTHUSIASM
~MRENTHUSIASM
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.