Hong Kong Preliminary Selection Contest

Note that $f\left(\frac{1}{x}\right)=\frac{15}{\frac{1}{x}+1}+\frac{16}{\frac{1}{x^2}+1}-\frac{17}{\frac{1}{x^3}+1}=\frac{15x}{x+1}+\frac{16x^2}{x^2+1}-\frac{17x^3}{x^3+1}$. Hence we have $$\begin{array}{rl}f(x)+f\left(\frac{1}{x}\right)& =15\left(\frac{1}{x+1}+\frac{x}{x+1}\right)+16\left(\frac{1}{x^2+1}+\frac{x^2}{x^2+1}\right)\\ & -17\left(\frac{1}{x^3+1}+\frac{x^3}{x^3+1}\right)\\ & =15+16-17=14.\end{array}$$ Since $\tan 15°=\frac{1}{\tan 75°}$, $\tan 30°=\frac{1}{\tan 60°}$ and $\tan 45°=1$, it follows that the value of the expression in the question is $14+14+f(1)=14+14+\frac{15}{2}+\frac{16}{2}-\frac{17}{2}=35$.