Difference between revisions of "2024 AMC 10A Problems/Problem 23"
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~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso], megaboy6679 (formatting), Technodoggo (LaTeX optimization/clarity adjustments) | ~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso], megaboy6679 (formatting), Technodoggo (LaTeX optimization/clarity adjustments) | ||
+ | |||
+ | ==Solution 6== $(Elimination) (I need help with formatting) | ||
+ | |||
+ | To solve the problem, we systematically test the options using elimination: | ||
+ | |||
+ | \subsection*{Step 1: Testing positive values} | ||
+ | We begin by testing three positive values, but none satisfy the equation when there is a plus sign. For example, | ||
+ | From this observation, we conclude that the answer cannot be | ||
+ | |||
+ | \subsection*{Step 2: Testing option | ||
+ | Option | ||
+ | \[ | ||
+ | a + b + c = -11 | ||
+ | \] | ||
+ | This sum is too large. Furthermore, if all three numbers are negative, the solution still fails. For example, testing | ||
+ | Thus, we rule out | ||
+ | |||
+ | \subsection*{Step 3: Testing options | ||
+ | For option | ||
+ | \[ | ||
+ | 247 - 284 = -37 | ||
+ | \] | ||
+ | Testing values such as | ||
+ | Therefore, | ||
+ | |||
+ | \subsection*{Step 4: Verifying | ||
+ | Finally, we test option | ||
+ | Thus, the correct answer is: | ||
+ | \[ | ||
+ | \boxed{\textbf{(D) } 276} | ||
+ | \] | ||
+ | |||
+ | \end{document} | ||
+ | |||
+ | ~pimathmonkey (explanation), ____ (formatting LaTex) | ||
== Video Solution by Power Solve == | == Video Solution by Power Solve == |
Revision as of 12:59, 17 November 2024
- The following problem is from both the 2024 AMC 10A #23 and 2024 AMC 12A #17, so both problems redirect to this page.
Contents
[hide]Problem
Integers , , and satisfy , , and . What is ?
Solution 1
Subtracting the first two equations yields . Notice that both factors are integers, so could equal one of and . We consider each case separately:
For , from the second equation, we see that . Then , which is not possible as is an integer, so this case is invalid.
For , we have and , which by experimentation on the factors of has no solution, so this is also invalid.
For , we have and , which by experimentation on the factors of has no solution, so this is also invalid.
Thus, we must have , so and . Thus , so . We can simply trial and error this to find that so then . The answer is then .
~eevee9406
minor edits by Lord_Erty09
Solution 2
Adding up first two equations:
Subtracting equation 1 from equation 2:
Which implies that from
Giving us that
Therefore,
~lptoggled
Solution 3 (Guess and check)
The idea is that you could guess values for , since then and are factors of . The important thing to realize is that , , and are all negative. Then, this can be solved in a few minutes, giving the solution , which gives the answer ~andliu766
Solution 4
Note that , and the only possible pair of results that yields this is and , so .
Therefore,
~luckuso, yuvag, Technodoggo (LaTeX credits to the latter two and editing to the latter)
Solution 5
There are ordered pairs of : , , .
However, only the last ordered pair meets all three equations.
Therefore,
~luckuso, megaboy6679 (formatting), Technodoggo (LaTeX optimization/clarity adjustments)
==Solution 6== $(Elimination) (I need help with formatting)
To solve the problem, we systematically test the options using elimination:
\subsection*{Step 1: Testing positive values}
We begin by testing three positive values, but none satisfy the equation when there is a plus sign. For example,
\subsection*{Step 2: Testing option
\subsection*{Step 3: Testing options
\subsection*{Step 4: Verifying
\end{document}
~pimathmonkey (explanation), ____ (formatting LaTex)
Video Solution by Power Solve
https://www.youtube.com/watch?v=LNYzBhf3Ke0
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=6SQ74nt3ynw
See also
2024 AMC 10A (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 AMC 10 Problems and Solutions |
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.