Difference between revisions of "2021 AMC 12A Problems/Problem 8"
MRENTHUSIASM (talk | contribs) (1. Fixed the code of the table, so the subscripts are really showing. 2. Minor revisions. Recall that we have to be cautious in the use of equal signs.) |
MRENTHUSIASM (talk | contribs) (→Solution) |
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\textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots | \textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots | ||
\end{array}</cmath> | \end{array}</cmath> | ||
− | Note that <math>(D_7,D_8,D_9)</math> | + | Note that <math>(D_7,D_8,D_9)</math> and <math>(D_0,D_1,D_2)</math> have the same parities, so the parity is periodic with period <math>7.</math> Since the remainders of <math>(2021\div7,2022\div7,2023\div7)</math> are <math>(5,6,7),</math> we conclude that <math>(D_{2021},D_{2022},D_{2023})</math> and <math>(D_5,D_6,D_7)</math> have the same parities, namely <math>\boxed{\textbf{(C) }(E,O,E)}.</math> |
~JHawk0224 ~MRENTHUSIASM | ~JHawk0224 ~MRENTHUSIASM |
Revision as of 07:46, 29 September 2021
Contents
Problem
A sequence of numbers is defined by and for . What are the parities (evenness or oddness) of the triple of numbers , where denotes even and denotes odd?
Solution
We construct the following table: Note that and have the same parities, so the parity is periodic with period Since the remainders of are we conclude that and have the same parities, namely
~JHawk0224 ~MRENTHUSIASM
Video Solution by Aaron He (Finding Cycles)
https://www.youtube.com/watch?v=xTGDKBthWsw&t=7m43s
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn (Using Parity and Pattern Finding)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/cckGBU2x1zg?t=227
~IceMatrix
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.