55rd Ukrainian National Mathematical Olympiad - Fourth Round

Denote blue coins by $B_1,\dots ,B_5$. Firstly, weigh 3 yellow and 3 blue coins $B_1,B_2,B_3$: $$1){\text{Ж}}_1+{\text{Ж}}_2+{\text{Ж}}_3?B_1+B_2+B_3.$$

$$1a){\text{Ж}}_1+{\text{Ж}}_2+{\text{Ж}}_3=B_1+B_2+B_3.$$ Fake coins are among $B_1,B_2,B_3$, or among $B_4$ and $B_5$. In a group where fakes are, all coins have different weight.

2) $B_1?B_2$. 2a) B_1 = B_2. It means that these coins are authentic, so $B_4$ and $B_5$ are fake. We only have to compare their weight. 2b) B_1 > B_2. Consider three possible cases. $B_1$ is heavier, $B_2$ lighter, or $B_1$ is heavier, $B_2$ authentic, or $B_1$ authentic, $B_2$ lighter. Compare total weight of coins $B_1, B_2$ and $\text{Ж}_1, \text{Ж}_2$ (both authentic). In the first case they are equal, in the second $B_1, B_2$ heavier, in the third - lighter than $\text{Ж}_1, \text{Ж}_2$, so we get answers to all questions after the third weighing: 3) \text{Ж}_1 + \text{Ж}_2 \ ? \ B_1 + B_2. 2c) B_1 < B_2. 16) \text{Ж}_1 + \text{Ж}_2 + \text{Ж}_3 > B_1 + B_2 + B_3. Among $B_1, B_2, B_3$ there is a lighter fake coin, and among $B_4, B_5$ - a heavier one. Then we use the following weighings: 2) B_1 \ ? \ B_2 \text{ and } 3) B_4 \ ? \ B_5. $$Theendinthiscaseisclear.$$ 1c) \text{Ж}_1 + \text{Ж}_2 + \text{Ж}_3 < B_1 + B_2 + B_3. $$Thiscaseissimilarto1b),butnowamong$B_1, B_2, B_3$ there is a heavier fake coin.