Difference between revisions of "2024 AMC 10B Problems/Problem 24"

Revision as of 13:08, 14 November 2024

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?

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

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

Solution 2 (Specific)

Take everything modulo 8 and re-write the entire fraction with denominator 8. This means that we're going to transform the fraction as follows : $P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}$ becomes $P(m)=\frac{4m + 2m^2 + m^4 + m^8}{8}$ And in order for $P(m)$ to be an integer, it's important to note that $4m + 2m^2 + m^4 + m^8$ must be congruent to 0 modulo 8. Moreover, we know that $2022 \equiv 6 \pmod 8, 2023 \equiv 7 \pmod 8, 2024 \equiv 0 \pmod 8, 2025 \equiv 1 \pmod 8$ . We can verify it by taking everything modulo 8 :

If $m = 2022$ , then $4(6) + 2(4) + 0 + 0 = 24 + 8 = 32 \equiv 0 \pmod 8$ -> TRUE If $m = 2023$ , then $4(7) + 2(1) + 1 + 1 = 28 + 2 + 1 + 1 = 32 \equiv 0 \pmod 8$ -> TRUE If $m = 2024$ , then it is obvious that the entire expression is divisible by 8. Therefore, it is true. If $m = 2025$ , then $2025 \equiv 1 \pmod 8$ . Therefore, $4(1) + 2(1) + 1 + 1 = 8 \equiv 0 \pmod 8$ -> TRUE Therefore, there are $\boxed{\textbf{(E) }4}$ possible values.

Addendum for certain China test papers : Note that $2026 \equiv 2 \pmod 8$ . Therefore, taking everything modulo 8, whilst still maintaining the original expression, gives $4(2) + 2(4) + 0 + 0 = 16 \equiv 0 \pmod 8$ . This is true.

Therefore, there are $\boxed{\textbf{(E) }5}$ possible values. ~elpianista227

Remark

On certain versions of the AMC in China, the problem was restated as follows:

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

By identical reasoning, each term of $P$ is an integer, since $2026$ is even.

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

~iHateGeometry, countmath1

Video Solution 1 by Pi Academy (In Less Than 2 Mins ⚡🚀)

https://youtu.be/Xn1JLzT7mW4?feature=shared

~ Pi Academy

@@ Line 23: / Line 23: @@
 <cmath>P(m)=\frac{4m + 2m^2 + m^4 + m^8}{8}</cmath>
 And in order for <math>P(m)</math> to be an integer, it's important to note that <math>4m + 2m^2 + m^4 + m^8</math> must be congruent to 0 modulo 8.
-Moreover, we know that <math>2022 \equiv 6 (mod 8), 2023 \equiv 7 (mod 8), 2024 \equiv 0 (mod 8), 2025 \equiv 1 (mod 8)</math>. We can verify it by taking everything modulo 8 :
+Moreover, we know that <math>2022 \equiv 6 \pmod 8, 2023 \equiv 7 \pmod 8, 2024 \equiv 0 \pmod 8, 2025 \equiv 1 \pmod 8</math>. We can verify it by taking everything modulo 8 :
-If <math>m = 2022</math>, then <math>4(6) + 2(4) + 0 + 0 = 24 + 8 = 32 \equiv 0 (mod 8)</math> -> TRUE
+If <math>m = 2022</math>, then <math>4(6) + 2(4) + 0 + 0 = 24 + 8 = 32 \equiv 0 \pmod 8</math> -> TRUE
-If <math>m = 2023</math>, then <math>4(7) + 2(1) + 1 + 1 = 28 + 2 + 1 + 1 = 32 \equiv 0 (mod 8)</math> -> TRUE
+If <math>m = 2023</math>, then <math>4(7) + 2(1) + 1 + 1 = 28 + 2 + 1 + 1 = 32 \equiv 0 \pmod 8</math> -> TRUE
 If <math>m = 2024</math>, then it is obvious that the entire expression is divisible by 8. Therefore, it is true.
-If <math>m = 2025</math>, then <math>2025 \equiv 1 (mod 8)</math>. Therefore, <math>4(1) + 2(1) + 1 + 1 = 8 \equiv 0 (mod 8)</math> -> TRUE
+If <math>m = 2025</math>, then <math>2025 \equiv 1 \pmod 8</math>. Therefore, <math>4(1) + 2(1) + 1 + 1 = 8 \equiv 0 \pmod 8</math> -> TRUE
 Therefore, there are <math>\boxed{\textbf{(E) }4}</math> possible values.
 Addendum for certain China test papers :
-Note that <math>2026 \equiv 2 (mod 8)</math>. Therefore, taking everything modulo 8, whilst still maintaining the original expression, gives <math>4(2) + 2(4) + 0 + 0 = 16 \equiv 0 (mod 8)</math>. This is true.
+Note that <math>2026 \equiv 2 \pmod 8</math>. Therefore, taking everything modulo 8, whilst still maintaining the original expression, gives <math>4(2) + 2(4) + 0 + 0 = 16 \equiv 0 \pmod 8</math>. This is true.
 Therefore, there are <math>\boxed{\textbf{(E) }5}</math> possible values.

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
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