11. In a class election, each of the five candidates got a different number of votes. There were 36 votes cast in total. The winner got 12 votes. The candidate in last place got 4 votes.
How many votes did the candidate in second place get?
A 8
B 9
C 8 or 9
D 9 or 10
E 10
SolUtion
C

Let the votes cast for each candidate be $12, x, y, z$ and 4 with $12>x>y>z>4$. Since there were 36 votes cast in total, we have $x+y+z=20$.
Since $y>z>4$, and $x, y$ and $z$ are integers, the minimum value of $y+z$ is $6+5=11$ and hence the maximum value $x$ can be is 9 with an overall solution for $(x, y, z)$ in that case being $(9,6,5)$. Also, since $7+6+5=18<20$, the minimum value $x$ can take is 8 with an overall solution for $(x, y, z)$ in that case being $(8,7,5)$. Hence, although it is not possible to determine exactly how many votes the candidate in second place received, we do know they received either 8 or 9 votes.