Difference between revisions of "2024 AMC 10A Problems/Problem 23"

Latest revision as of 20:28, 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.

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)

Solution 6 (Elimination)

Before we start, keep in mind that the problem is asking for the sum $ab+bc+ac$. This is nothing but $100+87+60-a-b-c$, or $247-(a+b+c$).

To solve the problem, we systematically test the options using elimination:

Let's first check options A and B, since they only happen when a,b, and c sum to 35 or 0. We begin by testing three positive values, but none satisfy the equation when there is a plus sign. For example, $ (12, 8, 4) $ satisfies $ ab + c = 100 $, but does not satisfy $ bc + a = 87$, or $ ac + b = 60$. If $a+b+c=0$, then not all of the numbers can be positive or negative, so this would not work. From this observation, we conclude that the answer cannot be $ \textbf{A} $ or $ \textbf{B} $.

Now let's test the next option, option 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, as we get results that are too small. Thus, we eliminate option $ \textbf{C} $.

Finally, let's test the last two options: D and 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.

Once we have this answer, we still need to verify it by testing out numbers: Finally, we test option $ \textbf{D} $. Using $ ab + bc + ca = 276 $, we get that $a+b+c = -29$. Also note that a, b, and c all have to be different, because the sums from the three equations are all different. We want to get the three closest values of a, b, and c such that they are all different, and the sum $a+b+c = -29$. The values $ (-9, -12, -8) $ are the closest three numbers. When we try them, they satisfy all three equations. So, the correct answer is: $ab+ba+ac= -9*-12+-12*-8+-8*-9 = \boxed{\textbf{(D) }276}.$

~pimathmonkey

Video Solution by Power Solve

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

Video Solution by SpreadTheMathLove

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

@@ Line 12: / Line 12: @@
 </math>
-==Solution==
+==Solution 1==
 Subtracting the first two equations yields <math>(a-c)(b-1)=13</math>. Notice that both factors are integers, so <math>b-1</math> could equal one of <math>13,1,-1,-13</math> and <math>b=14,2,0,-12</math>. We consider each case separately:
-For <math>b=0</math>, from the second equation, we see that <math>a=87</math>. Then <math>80c=60</math>, which is not possible as <math>c</math> is an integer, so this case is invalid.
+For <math>b=0</math>, from the second equation, we see that <math>a=87</math>. Then <math>87c=60</math>, which is not possible as <math>c</math> is an integer, so this case is invalid.
 For <math>b=2</math>, we have <math>2c+a=87</math> and <math>ca=58</math>, which by experimentation on the factors of <math>58</math> has no solution, so this is also invalid.
@@ Line 23: / Line 23: @@
 Thus, we must have <math>b=-12</math>, so <math>a=12c+87</math> and <math>ca=72</math>. Thus <math>c(12c+87)=72</math>, so <math>c(4c+29)=24</math>. We can simply trial and error this to find that <math>c=-8</math> so then <math>a=-9</math>. The answer is then <math>(-9)(-12)+(-12)(-8)+(-8)(-9)=108+96+72=\boxed{\textbf{(D) }276}</math>.
 ~eevee9406
+ minor edits by Lord_Erty09
 ==Solution 2==
-<math>(a+c)(b+1)=187</math>
+Adding up first two equations: <cmath>(a+c)(b+1)=187</cmath>
-<math>b+1=±11,±17</math>
+<cmath>b+1=\pm 11,\pm 17</cmath>
-<math>b=-12,10,-18,16</math>
+<cmath>b=-12,10,-18,16</cmath>
-<math>(a-c)(b-1)=13</math>
-<math>b-1=±1,±13</math>
+Subtracting equation 1 from equation 2: <cmath>(a-c)(b-1)=13</cmath>
-<math>b=0,2,-12,14</math>
+<cmath>b-1=\pm 1,\pm 13</cmath>
-<math>\rightarrow b=-12</math>
+<cmath>b=0,2,-12,14</cmath>
-Which implies that <math>a+c=-17</math>
-Therefore, ab+ba+ac=100+87+60-(a+b+c)
+<cmath>\Rightarrow b=-12</cmath>
-<math>=\boxed{\text{(B) }276}</math>
+Which implies that <math>a+c=-17</math> from <math>(a+c)(b+1)=187</math>
+Giving us that <math>a+b+c=-29</math>
+Therefore, <math>ab+bc+ac=100+87+60-(a+b+c)=\boxed{\text{(D) }276}</math>
+~lptoggled
+== Solution 3 (Guess and check) ==
+The idea is that you could guess values for <math>c</math>, since then <math>a</math> and <math>b</math> are factors of <math>100 - c</math>. The important thing to realize is that <math>a</math>, <math>b</math>, and <math>c</math> are all negative. Then, this can be solved in a few minutes, giving the solution <math>(-9, -12, -8)</math>, which gives the answer <math>\boxed{\textbf{(D)} 276}</math>
+~andliu766
+==Solution 4==
+<cmath>\begin{align}
+ab + c &= 100 \\
+bc + a &= 87 \\
+ca + b &= 60
+\end{align}</cmath>
+<cmath>(1) + (2) \implies  ab + c +bc + a = (a+c)(b+1)=187\implies b+1=\pm 11,\pm 17</cmath>
+<cmath>(1) - (2) \implies ab + c - bc - a = (a-c)(b-1)=13\implies b-1=\pm 1,\pm 13</cmath>
+Note that <math>(b+1)-(b-1)=2</math>, and the only possible pair of results that yields this is <math>b-1=-13</math> and <math>b+1=-11</math>, so <math>a+c=-17</math>.
+Therefore,
+<cmath>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}.</cmath>
+~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso], yuvag, Technodoggo (LaTeX credits to the latter two and editing to the latter)
+==Solution 5==
+<cmath>\begin{align}
+ab + c &= 100 \\
+bc + a &= 87 \\
+ca + b &= 60
+\end{align}</cmath>
+\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 <math>3</math> ordered pairs of <math>(a,b,c)</math>: <math>(5,14,4)</math>, <math>(-3,-12,-3)</math>, <math>(-9,-12,-8)</math>.
+However, only the last ordered pair meets all three equations.
+Therefore, <math>ab+ba+ac= -9*-12+-12*-8+-8*-9 = \boxed{\textbf{(D) }276}.</math>
+~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso], megaboy6679 (formatting), Technodoggo (LaTeX optimization/clarity adjustments)
+==Solution 6 (Elimination)==
+Before we start, keep in mind that the problem is asking for the sum \(ab+bc+ac\). This is nothing but \(100+87+60-a-b-c\), or \(247-(a+b+c\)).
+To solve the problem, we systematically test the options using elimination:
+Let's first check options A and B, since they only happen when a,b, and c sum to 35 or 0.
+We begin by testing three positive values, but none satisfy the equation when there is a plus sign. For example, \( (12, 8, 4) \) satisfies \( ab + c = 100 \), but does not satisfy \( bc + a = 87\), or \( ac + b = 60\). If \(a+b+c=0\), then not all of the numbers can be positive or negative, so this would not work.
+From this observation, we conclude that the answer cannot be \( \textbf{A} \) or \( \textbf{B} \).
+Now let's test the next option, option 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, as we get results that are too small. Thus, we eliminate option \( \textbf{C} \).
+Finally, let's test the last two options: D and 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.
+Once we have this answer, we still need to verify it by testing out numbers:
+Finally, we test option \( \textbf{D} \). Using \( ab + bc + ca = 276 \), we get that \(a+b+c = -29\). Also note that a, b, and c all have to be different, because the sums from the three equations are all different. We want to get the three closest values of a, b, and c such that they are all different, and the sum \(a+b+c = -29\). The values \( (-9, -12, -8) \) are the closest three numbers. When we try them, they satisfy all three equations.
+So, the correct answer is:
+<math>ab+ba+ac= -9*-12+-12*-8+-8*-9 = \boxed{\textbf{(D) }276}.</math>
+~pimathmonkey
+== 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==
 {{AMC10 box|year=2024|ab=A|num-b=22|num-a=24}}
 {{AMC12 box|year=2024|ab=A|num-b=16|num-a=18}}
 {{MAA Notice}}

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