Question 67
Let $n$ be a positive integer and $a_1, \dots, a_{2n}$ be real numbers in $\left[-\frac{1}{2}, \frac{1}{2}\right]$. Leaving out any one of the numbers, the sum of the remaining $2n-1$ numbers is always an integer. Prove that $a_1 = \dots = a_{2n}$.