2025 AMC 8 Problem 16
Attempt Problem 16 of the 2025 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
16.
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?
Answer: C
Small hint:
Pair each number from to with the number larger
Big hint:
The chosen upper numbers correspond to the unchosen lower numbers
Video solution:
Click to load, then click again to play
Written solution:
Call the integers from to inclusive the lower range, and call the integers from to inclusive the higher range.
Each of the distinct numbers chosen from the lower range blocks out the number in the higher range that is exactly more than itself. There are only numbers in the higher range, so there are only numbers not yet blocked.
We need to choose distinct numbers from the higher range, so the numbers chosen from the higher range are precisely those which are not yet blocked. They are each exactly more than a not-chosen number in the lower range.
So, the sum of the distinct numbers chosen from the higher range is exactly more than the sum of the not-chosen numbers in the lower range.
The sum of all chosen numbers is therefore equal to plus the sum of all chosen and not-chosen numbers in the lower range
The sum of the numbers from to is so the answer is or choice C.
Problem 16 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2009 AMC 8
- 2010 AMC 8
- 2011 AMC 8
- 2012 AMC 8
- 2013 AMC 8
- 2014 AMC 8
- 2015 AMC 8
- 2016 AMC 8
- 2017 AMC 8
- 2018 AMC 8
- 2019 AMC 8
- 2020 AMC 8
- 2022 AMC 8
- 2023 AMC 8
- 2024 AMC 8
- 2026 AMC 8