55rd Ukrainian National Mathematical Olympiad - Third Round

At the first move, put $1+4+9$ on the left side and $2+5+7$ on the right side. If the total weights are equal, the counterfeit weight is among the other three. Then put on scales the combinations $3+4$ and $1+6$.

If $3+4=1+6$, they are genuine, hence, $8$ is counterfeit.

If $3+4<1+6$, then $3$ is counterfeit, because it is lighter and $4$ is known to be genuine. * Similarly, $3+4>1+6$ would imply that $6$ is counterfeit.

If the total weights at the first weighing are not equal, the counterfeit weight is among the three lighter ones. The second move involves putting one potentially counterfeit weight on each side and balancing them with genuine ones. For example, $1+5$ and $4+2$. Then we use the same arguments: the counterfeit one is where the total weight is less. If the scales are balanced, the unused weight is counterfeit.