$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$

Let $ a_n=1+\dfrac1{2^2}+\dfrac1{3^2}+\cdots+\dfrac1{n^2}$ Find $ \lim_{n\to\infty}a_n$

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .