Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?

The Fibonacci sequence is defined by $f_1 = 0, f_2 = 1$, and $f_{n+2} = f_{n+1}+f_n$ for $n\ge 1$. Prove that there is a strictly increasing arithmetic progression whose no term is in the Fibonacci sequence.

Given a finite sequence $x_{1,1}, x_{2,1}, \dots , x_{n,1}$ of integers $(n\ge 2)$, not all equal, define the sequences $x_{1,k}, \dots , x_{n,k}$ by \[ x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k})\quad\text{where }x_{n+1,k}=x_{1,k}.\] Show that if $n$ is odd, then not all $x_{j,k}$ are integers. Is this also true for even $n$?

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]