2023 AMC 12A Problems/Problem 12
Contents
Problem
What is the value of
Solution 1
To solve this problem, we will be using difference of cube, sum of squares and sum of arithmetic sequence formulas.
we could rewrite the second part as
Hence,
Adding everything up:
~lptoggled
Solution 2
Think about . Once we factor out , we get , which can be found using the sum of cubes formula, . Now think about . This is just the previous sum subtracted from the total sum of cubes. So now we have the two things we need to add. The sum of all the even cubes is . The sum of all cubes from to is . The sum of the odd cubes is then . Thus we get ~amcrunner
Solution 2 (a bit faster)
Using the same sum of cubes formula, we can rewrite as
~AoPSuser216
Solution 3
For any real numbers and , .
When , with the above formula, we will get .
Therefore,
~sqroot
Alternatively, to avoid the long sum,
Solution 4
We rewrite the sum as
-Benedict T (countmath1)
Solution 4 (a bit faster)
We see which is clearly a multiple of 9. The only answer choice which is a multiple of 9 is ~Ilaggo2432
Solution 5 (Bash)
Solution 6
Reduce all terms mod 9. This yields:
The only answer choice which is also ≡0 mod 9 is
Video Solution (easy to digest) by Power Solve
https://youtu.be/YXIH3UbLqK8?si=RZhSDIKjRNLrgVS5&t=1552
Video Solution
https://youtu.be/33Tz-bfKzmw ~Education, the Study of Everything
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 AMC 12 Problems and Solutions |
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