Difference between revisions of "2024 AMC 10B Problems/Problem 24"
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{{duplicate|[[2024 AMC 10B Problems/Problem 24|2024 AMC 10B #24]] and [[2024 AMC 12B Problems/Problem 18|2024 AMC 12B #18]]}} | {{duplicate|[[2024 AMC 10B Problems/Problem 24|2024 AMC 10B #24]] and [[2024 AMC 12B Problems/Problem 18|2024 AMC 12B #18]]}} | ||
− | ==Problem | + | ==Problem== |
− | + | Let | |
− | <math>\textbf{(A) } | + | <cmath>P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}</cmath> |
+ | How many of the values <math>P(2022)</math>, <math>P(2023)</math>, <math>P(2024)</math>, and <math>P(2025)</math> are integers? | ||
+ | |||
+ | <math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math> | ||
==Solution 1== | ==Solution 1== |
Revision as of 05:53, 14 November 2024
- The following problem is from both the 2024 AMC 10B #24 and 2024 AMC 12B #18, so both problems redirect to this page.
Contents
[hide]Problem
Let How many of the values , , , and are integers?
Solution 1
The first terms
so the answer is .
Solution 2
Define new sequence
A= and B =
Per characteristic equation, itself is also Fibonacci type sequence with starting item
then we can calculate the first 10 items using
so the answer is .
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 10 Problems and Solutions |
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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.