G1.2 Among 50 school teams joining the HKMO, no team answered all four questions correctly in the paper of a group event. If the first question was solved by 45 teams, the second by 40 teams, the third by 35 teams and the fourth by 30 teams. How many teams solved both the third and the fourth questions?
$\because$ No team answered all four questions correctly
$\therefore$ Each team can solve at most three questions.
The maximum number of solved questions $=50 \times 3=150$
The actual number of solved questions $=45+40+35+30=150$
$\therefore$ Each team can solve exactly three questions.
Number of teams that cannot solve the first question $=(50-45)$ teams $=5$ teams
$\Rightarrow$ These 5 teams can solve Q2, Q3 and Q4 but not Q1.
Number of teams that cannot solve the second question $=(50-40)$ teams $=10$ teams
$\Rightarrow$ These 10 teams can solve Q1, Q3 and Q4 but not Q2.
Number of teams that cannot solve the third question $=(50-35)$ teams $=15$ teams
$\Rightarrow$ These 15 teams can solve Q1, Q2 and Q4 but not Q3.
Number of teams that cannot solve the fourth question $=(50-30)$ teams $=20$ teams
$\Rightarrow$ These 20 teams can solve $\mathrm{Q} 1, \mathrm{Q} 2$ and $\mathrm{Q} 3$ but not $\mathrm{Q} 4$.
Number of school teams solved both the third and the fourth questions $=5+10=15$
Remark We cannot use the Venn diagram on the right with
explanation below:
$x=$ school teams that can solve $\mathrm{Q} 1$ but not $\mathrm{Q} 2, \mathrm{Q} 3$ nor $\mathrm{Q} 4$.
$y=$ school teams that can solve $\mathrm{Q} 1, \mathrm{Q} 2$ but not $\mathrm{Q} 3$ nor $\mathrm{Q} 4$.
$z=$ school teams that can solve all four questions.
$t=$ school teams that can solve $\mathrm{Q} 1, \mathrm{Q} 3$ and $\mathrm{Q} 4$ but not $\mathrm{Q} 2$
However, we could not find any part in Venn diagram representing school teams that can solve Q1, Q4 but not Q2 nor Q3 !!!