2019 AMC 8 Problems/Problem 23
Contents
Problem 23
After Euclid High School's last basketball game, it was determined that 
of the team's points were scored by Alexa and 
were scored by Brittany. Chelsea scored 
points. None of the other 
team members scored more than 
points What was the total number of points scored by the other team members?
Solution 1
Since 
and 
are integers, we have 
. We see that the number of points scored by the other team members is less than or equal to 
and greater than or equal to 
. We let the total number of points be 
and the total number of points scored by the other team members, which means that 
, which means 
. The only value of that satisfies all conditions listed is 
, so 
. - juliankuang
Solution 2
Starting from the above equation 
where is the total number of points scored and 
is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation 
, or 
. Since is necessarily divisible by 28, let 
where 
and divide by 28 to obtain 
. Then it is easy to see 
(
) is the only candidate, giving . -scrabbler94
See Also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
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