3. From 80 identical Lego parts, several figures were assembled, with the number of parts used in all figures being different. For the manufacture of the three smallest figures, 14 parts were used, and in the three largest, 43 were used in total. How many figures were assembled? How many parts are in the largest figure?
Answer: 8 figurines, 16 parts.

Solution. Let the number of parts in the figurines be denoted by $a_{1}43$, so $a_{n-2} \leqslant 13$.

Remove the three largest and three smallest figurines. In the remaining figurines, there will be $80-14-$ $43=23$ parts, and each will have between 7 and 12 parts. One figurine is clearly insufficient, and three would be too many $(7+8+9=24)$. Therefore, 23 parts form 2 figurines. This is possible, and in only one way: $23=11+12$. We have $43=13+14+16$ - the only decomposition with $a_{6} \geqslant 13$.

Criteria. Only the answer - 0 points. Only correct estimates for $a_{3}$ and $a_{n-2}-3$ points. Only a justified answer for the number of figurines - 5 points. Complete solution - 7 points.