IYMC 2022 Qualification

Problem A: What are the roots of the function $f(x)=(\log(3^{x})-2\log(3))\cdot(x^2-1)$ with $x\in\mathbb{R}?$ $$f(x)=(x-2)(\log(3))\cdot(x^{2}-1)$$$$x=-1,1,2$$ Problem B: Find the values of the following infinite sum $$1+\frac{3}{\pi}+\frac{3}{\pi^2}+\frac{3}{\pi^3}+\frac{3}{\pi^4}+\frac{3}{\pi^5}+\cdots$$ We can rewrite the sum as $$1+\frac{3}{\pi}\bigg(1+\frac{1}{\pi}+\frac{1}{\pi^2}+\frac{1}{\pi^3}+\frac{1}{\pi^4}+\cdots\bigg)$$ Using the formula for the sum of geometric progressions and noting that $\displaystyle\frac{1}{\pi}<1,$ we can write $$1+\frac{3}{\pi}\bigg(\frac{1}{1-1/\pi}\bigg)=\frac{\pi+2}{\pi-1}$$