Rioplatense Mathematical Olympiad

The answer is no. We will show that, at all times, the amount of liters of orange juice in any bottle can be represented as $\frac{m}{3^k}$ for some nonnegative integers $m,k$. This is clearly true at the beginning of the process. On each move, we pick one bottle that has $\frac{m_1}{3^{k_1}}$ and add $\frac{m_1}{3^{k_1+1}}$ to three bottles. Assuming one of these bottles previously contained $\frac{m_2}{3^{k_2}}$ litres of orange juice, after our move that bottle ends up with $\frac{m_2}{3^{k_2}}+\frac{m_1}{3^{k_1+1}}=\frac{m_2{3}^{k_1+1}+m_1{3}^{k_2}}{3^{k_1+k_2+1}}$, proving it will still be of the desired form. Since $\frac{1}{10}$ cannot be represented in such a way, because 10 is not a power of 3, this proves that it is not possible that all bottles end up with the same amount of orange juice.