National Olympiad of Argentina

The answer is $126$. Let there be $a_i$ girls and $b_i$ boys in grade $i$, $1\le i\le 5$. Consider a $2\times 5$ table with $a_1,\dots ,a_5$ in the first row and $b_1,\dots ,b_5$ in the second row. Mark the smaller of the numbers $a_i,b_i$ for each $i$. The number of teams that can be formed is the sum of the five marked numbers. At least three marked numbers are in the same row. Suppose for instance that $a_1,a_2,a_3$ are marked. Since $b_4\ge 19$, $b_5\ge 19$ and $a_4+b_4=a_5+b_5=100$, each of $a_4$ and $a_5$ is at most $100-19=81$. Because $a_1+a_2+a_3+a_4+a_5=250$, it follows that $a_1+a_2+a_3=250-(a_4+a_5)\ge 250-2\cdot 81=88$.

Due to $a_i\le b_i$, $1\le i\le 3$, the number of teams in grades 1, 2, 3 is $a_1+a_2+a_3$, hence it is at least $88$. Also at least $19$ teams can be formed in each of grades 4 and 5. So $88+2\cdot 19=126$ teams can be formed always. The example $(a_1,a_2,a_3,a_4,a_5)=(29,29,30,81,81)$, $(b_1,b_2,b_3,b_4,b_5)=(71,71,70,19,19)$ shows that $126$ is the greatest number in question.