Question 1014123
The total of the three scores must be {{{3(5+3+1)=3*9}}} ,
which makes the average score {{{3*9/3=9}} .
So the highest scorer must score more than {{{9}}} .
The highest score must also be odd, because it is the sum of three odd numbers.
Could it be {{{11}}} ?
 
There are 10 possible scores:
{{{1+1+1=3}}}
{{{1+1+3=5}}}
{{{1+1+5=7}}}
{{{1+3+3=7}}}
{{{1+3+5=9}}}
{{{1+5+5=11}}}
{{{3+3+3=9}}}
{{{3+3+5=11}}}
{{{3+5+5=13}}}
{{{5+5+5=15}}}
If Beth scores were 1, 5, and 5 (in any order),
the remaining scores would be 1, 1, 3, 3, 3, and 5.
There is at least one way in which Amanda and Sarah could have earned those scores,
and still have lower total score:
{{{matrix(4,5,
name, 1st, 2nd, 3rd, total,
Beth, 1,5,5,11,
Amanda,5,3,1,9,
Sarah,3,1,3,7)}}} , so Beth could have a score of 11, and still be the highest scorer.
The smallest total score that Beth can have is {{{highlight(11)}}} .