Difference between revisions of "2025 AMC 8 Problems/Problem 16"
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Five distinct integers from <math>1</math> to <math>10</math> are chosen, and five distinct integers from <math>11</math> to <math>20</math> are chosen. No two numbers differ by exactly <math>10</math>. What is the sum of the ten chosen numbers? | Five distinct integers from <math>1</math> to <math>10</math> are chosen, and five distinct integers from <math>11</math> to <math>20</math> are chosen. No two numbers differ by exactly <math>10</math>. What is the sum of the ten chosen numbers? | ||
Latest revision as of 19:15, 30 January 2025
Contents
Problem
Five distinct integers from 
to 
are chosen, and five distinct integers from 
to 
are chosen. No two numbers differ by exactly . What is the sum of the ten chosen numbers?
Solution
Note that for no two numbers to differ by , every number chosen must have a different units digit. To make computations easier, we can choose 
from the first group and 
from the second group. Then the sum evaluates to 
.
~Soupboy0
Similar solution
One efficient method is to quickly add 
, which is 
. Then because you took 
in total away from , you add . 
.
~Bepin999
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
