Difference between revisions of "2016 AIME II Problems/Problem 6"
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| − | Note that all the odd coefficients have an odd number of odd degree terms multiplied together, and all the even coefficients have an even number of odd degree terms multiplied together. Since every odd degree term is negative, and every even degree term is positive, the sum is just equal to <math>Q(-1)=P(-1)^{5}=(\dfrac{3}{2})^{5}=\dfrac{243}{32}</math>, so the desired answer is <math>243+32=\boxed{275}</math>. | + | Note that all the odd coefficients have an odd number of odd degree terms multiplied together, and all the even coefficients have an even number of odd degree terms multiplied together. Since every odd degree term is negative, and every even degree term is positive, the sum is just equal to <math>Q(-1)=P(-1)^{5}=\left( \dfrac{3}{2}\right)^{5}=\dfrac{243}{32}</math>, so the desired answer is <math>243+32=\boxed{275}</math>. |
Solution by Shaddoll | Solution by Shaddoll | ||
Revision as of 19:40, 17 March 2016
For polynomial 
, define
.
Then 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Note that all the odd coefficients have an odd number of odd degree terms multiplied together, and all the even coefficients have an even number of odd degree terms multiplied together. Since every odd degree term is negative, and every even degree term is positive, the sum is just equal to 
, so the desired answer is 
.
Solution by Shaddoll
