# Difference between revisions of "2021 April MIMC 10 Problems/Problem 17"

Cellsecret (talk | contribs) (→Solution) |
Cellsecret (talk | contribs) (→Solution) |
||

Line 4: | Line 4: | ||

==Solution== | ==Solution== | ||

− | |||

<cmath>\sum_{k=0}^{60} {60 \choose k}</cmath> can be expressed as <math>2^{60}</math>, and <math>60 \choose 0</math> is equal to <math>1</math>. Therefore, we can simplify the original expression into <math>2^{60}-1+2^{59}-1+...+2^3-1-2^{10}=2^{60}+2^{59}+...+2^{3}+2^3-58-1024=2^{61}-(8+58+1024)=2^{61}-1090</math>. The expression that the answer wants would be <math>2^2\cdot(2\cdot 61+2\cdot2+1090)=4\cdot 1216=\fbox{\textbf{(D)} 4864}</math>. | <cmath>\sum_{k=0}^{60} {60 \choose k}</cmath> can be expressed as <math>2^{60}</math>, and <math>60 \choose 0</math> is equal to <math>1</math>. Therefore, we can simplify the original expression into <math>2^{60}-1+2^{59}-1+...+2^3-1-2^{10}=2^{60}+2^{59}+...+2^{3}+2^3-58-1024=2^{61}-(8+58+1024)=2^{61}-1090</math>. The expression that the answer wants would be <math>2^2\cdot(2\cdot 61+2\cdot2+1090)=4\cdot 1216=\fbox{\textbf{(D)} 4864}</math>. |

## Revision as of 13:50, 26 April 2021

The following expression can be expressed as which both and are relatively prime positive integers. Find .

## Solution

can be expressed as , and is equal to . Therefore, we can simplify the original expression into . The expression that the answer wants would be .