2016
Problem - 3008
For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\displaystyle\sum_{i=0}^{50} a_ix^{i}$. Find the value of $\displaystyle\sum_{i=0}^{50} |a_i|$.
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 $$Q(-1)=P(-1)^{5}=\left( \dfrac{3}{2}\right)^{5}=\boxed{\dfrac{243}{32}}$$