2024 AMC 10B Problems/Problem 24

Problem

Let $P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}$ How many of the values $P(2022)$ , $P(2023)$ , $P(2024)$ , and $P(2025)$ are integers?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Certain China test papers: Let $P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}$ How many of the values $P(2022)$ , $P(2023)$ , $P(2024)$ , $P(2025)$ and $P(2026)$ are integers?

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$

Solution (The simplest way)

First, we know that $P(2022)$ and $P(2024)$ must be integers since they are both divisible by $2$ .

Then Let’s consider the remaining two numbers. Since they are not divisible by $2$ , the result of the first term must be a certain number $+\frac{1}{2}$ , and the result of the second term must be a certain number $+\frac{1}{4}$ . Similarly, the remaining two terms must each be $\frac{1}{8}$ . Their sum is $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} = 1$ , so $P(2023)$ and $P(2025)$ are also integers.

Therefore, the answer is $\boxed{\textbf{(E) }4}$ .

~Athmyx

Certain China test papers:

As explained above, numbers $P(2022)$ to $P(2025)$ are integers. The difference is that there is a new number, $P(2026)$ . Since it is divisible by 2, we can see that it is also an integer.

Therefore, the answer is $\boxed{\textbf{(E) }5}$ .

~iHateGeometry

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

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2024 AMC 10B Problems/Problem 24

Problem

Solution (The simplest way)

See also