2025 AMC 12A Problems/Problem 21: Difference between revisions

Latest revision as of 01:21, 11 November 2025

Problem 21

There is a unique ordered triple $(a,k,m)$ of nonnegative integers such that $\frac{4^a + 4^{a+k}+4^{a+2k}+\cdots + 4^{a+mk}}{2^a + 2^{a+k} + 2^{a+2k}+ \cdots + 2^{a+mk}} = 964.$ What is $a+k+m$ ?

$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C)} 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

Solution 1

The numerator can be written as $2^{2a} +2^{2a+2k}+...+2^{2a+2mk}$ , which is actually a sum of geometric series. This can be expressed as $2^{2a} \cdot \frac{1-2^{2k(m+1)}}{1-2^{2k}}$ . The denominator in the same way can be expressed as $2^a \cdot \frac{1-2^{k(m+1)}}{1-2^k}$ .

Doing some algebra on the top and bottom we get:

\begin{equation} 2^a \cdot \frac{1+2^{k(m+1)}}{1+2^k} = 964 \end{equation}

The prime factorization of $964$ is $241 \cdot 2^2$ .

Equating $2^a = 2^2$ , we get $a = 2$ .

Next since $241$ is a prime number we equate the latter half of the product to $241$ . \begin{equation} \frac{1+2^{k(m+1)}}{1+2^k} = 241 \end{equation} Doing some algebra we get, that $2^k(2^{km}-241) = 240$

The closest power of $2$ to $241$ is $256$ which is $2^8$ . So setting $km = 8$ , we get $2^8-241 = 15$

$2^k(15) = 240$ , $2^k = 16$ , $k = 4$ .

This we know that if $k = 4$ , then $4m = 8$ , so $m = 2$ .

Finally, we conclude that $a = 2$ , $m = 2$ , and $k = 4$ .

4+2+2 = 8, so the answer is $\fbox{A}$

~MATHEMATICIAN635 ~Minor Edits by MALICIOUSFISH23

Actually when it comes to $2^k(2^{km}-241) = 240$ ; we can directly observe the answer by discovering $2^k$ is even and $2^{km}-241$ must be odd. So, the only available divisors of 240 is 16 and 15. And we can have $m = 2$ , and $k = 4$ .

~DRA777

Video Solution 1 by OmegaLearn

https://youtu.be/Cx8BScqcx6U

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=Y9sG6vZPDQo

@@ Line 31: / Line 31: @@
 Finally, we conclude that <imath>a = 2</imath>, <imath>m = 2</imath>, and <imath>k = 4</imath>.
-+2+2 = 8, so the answer is A
++2+2 = 8, so the answer is <imath>\fbox{A}</imath>
 ~MATHEMATICIAN635
@@ Line 44: / Line 44: @@
 ==Video Solution 1 by OmegaLearn==
 https://youtu.be/Cx8BScqcx6U
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=Y9sG6vZPDQo
 ==See Also==

2025 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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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