62nd Ukrainian National Mathematical Olympiad

We denote the numbers in the circle by $a_1,a_2,\dots ,a_n$. Choose the largest among them (or any of them, if there are several such numbers). Without loss of generality, let this number be $a_1$. From the condition of the problem we have that $$a_1+a_2=a_n+a_3,a_1+a_n=a_2+a_{n-1}\Rightarrow 2a_1=a_3+a_{n-1}.$$ Since $a_1$ is the largest number, the last equality implies that $a_1=a_3=a_{n-1}$: that is, every second number is the largest, if we continue with the same reasoning. If $n$ is odd, then all the numbers will be equal.

For even numbers $n$ it is enough to consider the following example with not all equal numbers: $1,0,1,0,\dots ,1,0$.