2024 AMC 10A Problems/Problem 23

The following problem is from both the 2024 AMC 10A #23 and 2024 AMC 12A #17, so both problems redirect to this page.

Problem

Integers $a$ , $b$ , and $c$ satisfy $ab + c = 100$ , $bc + a = 87$ , and $ca + b = 60$ . What is $ab + bc + ca$ ?

$\textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad$

Solution 1

Subtracting the first two equations yields $(a-c)(b-1)=13$ . Notice that both factors are integers, so $b-1$ could equal one of $13,1,-1,-13$ and $b=14,2,0,-12$ . We consider each case separately:

For $b=0$ , from the second equation, we see that $a=87$ . Then $87c=60$ , which is not possible as $c$ is an integer, so this case is invalid.

For $b=2$ , we have $2c+a=87$ and $ca=58$ , which by experimentation on the factors of $58$ has no solution, so this is also invalid.

For $b=14$ , we have $14c+a=87$ and $ca=46$ , which by experimentation on the factors of $46$ has no solution, so this is also invalid.

Thus, we must have $b=-12$ , so $a=12c+87$ and $ca=72$ . Thus $c(12c+87)=72$ , so $c(4c+29)=24$ . We can simply trial and error this to find that $c=-8$ so then $a=-9$ . The answer is then $(-9)(-12)+(-12)(-8)+(-8)(-9)=108+96+72=\boxed{\textbf{(D) }276}$ .

~eevee9406

minor edits by Lord_Erty09

Solution 2

Adding up first two equations: $(a+c)(b+1)=187$ $b+1=\pm 11,\pm 17$ $b=-12,10,-18,16$

Subtracting equation 1 from equation 2: $(a-c)(b-1)=13$ $b-1=\pm 1,\pm 13$ $b=0,2,-12,14$

$\Rightarrow b=-12$

Which implies that $a+c=-17$ from $(a+c)(b+1)=187$

Giving us that $a+b+c=-29$

Therefore, $ab+bc+ac=100+87+60-(a+b+c)=\boxed{\text{(D) }276}$

~lptoggled

Solution 3 (Guess and check)

The idea is that you could guess values for $c$ , since then $a$ and $b$ are factors of $100 - c$ . The important thing to realize is that $a$ , $b$ , and $c$ are all negative. Then, this can be solved in a few minutes, giving the solution $(-9, -12, -8)$ , which gives the answer $\boxed{\textbf{(D)} 276}$ ~andliu766

Solution 4

$\begin{align} ab + c &= 100 \\ bc + a &= 87 \\ ca + b &= 60 \end{align}$

$(1) + (2) \implies ab + c +bc + a = (a+c)(b+1)=187\implies b+1=\pm 11,\pm 17$

$(1) - (2) \implies ab + c - bc - a = (a-c)(b-1)=13\implies b-1=\pm 1,\pm 13$

Note that $(b+1)-(b-1)=2$ , and the only possible pair of results that yields this is $b-1=-13$ and $b+1=-11$ , so $a+c=-17$ .

Therefore,

$ab+ba+ac=ab + c +bc + a + ca + b -(a+b+c) = (1)+(2)+(3) - (a+b+c) = 100+87+60-(a+b+c)=\boxed{\textbf{(D) }276}.$ ~luckuso, yuvag, Technodoggo (LaTeX credits to the latter two and editing to the latter)

Solution 5

$\begin{align} ab + c &= 100 \\ bc + a &= 87 \\ ca + b &= 60 \end{align}$

\begin{align*} (1) - (2) \implies ab + c -bc - a &=(a-c)(b-1)=13 \\ (2) - (3) \implies bc + a -ca - b &=(b-a)(c-1)=27 \\ (3) - (1) \implies ca + b -ab - c &=(c-b)(a-1)=-40 \end{align*}

There are $3$ ordered pairs of $(a,b,c)$ : $(5,14,4)$ , $(-3,-12,-3)$ , $(-9,-12,-8)$ .

However, only the last ordered pair meets all three equations.

Therefore, $ab+ba+ac= -9*-12+-12*-8+-8*-9 = \boxed{\textbf{(D) }276}.$

~luckuso, megaboy6679 (formatting), Technodoggo (LaTeX optimization/clarity adjustments)

\documentclass[12pt]{article} \usepackage{amsmath, amssymb, amsfonts} \usepackage[margin=1in]{geometry} % Adjusts the page margins for better formatting \usepackage{lmodern} % Enhances font appearance

\begin{document}

\section*{Solution 6: Elimination (Last Approach)}

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, $ (12, 8, 4) $ does not come close to working. From this observation, we conclude that the answer cannot be $ \textbf{A} $ or $ \textbf{B} $.

\subsection*{Step 2: Testing option $ \textbf{C} $} Option $ \textbf{C} $ states $ ab + bc + ca = 258 $. If true, then: \[ a + b + c = -11 \] This sum is too large. Furthermore, if all three numbers are negative, the solution still fails. For example, testing $ (-4, -5, -2) $ confirms the equation is not satisfied. Thus, we rule out $ \textbf{C} $.

\subsection*{Step 3: Testing options $ \textbf{D} $ and $ \textbf{E} $} For option $ \textbf{E} $, the sum $ a + b + c $ would be: \[ 247 - 284 = -37 \] Testing values such as $ (-11, -12, -14) $, the resulting sums $ ab + c $, $ bc + a $, and $ ac + b $ are far too large to satisfy the equation. Therefore, $ \textbf{E} $ is also eliminated.

\subsection*{Step 4: Verifying $ \textbf{D} $} Finally, we test option $ \textbf{D} $. Using $ ab + bc + ca = 276 $, the values $ (-9, -12, -8) $ satisfy the equation. Thus, the correct answer is: \[ \boxed{\textbf{(D) } 276} \]

\end{document}

Video Solution by Power Solve

https://www.youtube.com/watch?v=LNYzBhf3Ke0

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=6SQ74nt3ynw

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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All AMC 10 Problems and Solutions

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AMC 12 Problems and Solutions

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