AMC 10A/12A 2024 (Problem 23/17)Integers aaa, bbb, and ccc satisfy ab+c=100ab+c=100ab+c=100, bc+a=87bc+a=87bc+a=87, and ca+b=60ca+b=60ca+b=60. What is ab+bc+caab+bc+caab+bc+ca?(A) 212\text{(A)}\;212(A)212(B) 247\text{(B)}\;247(B)247(C) 258\text{(C)}\;258(C)258(D) 276\text{(D)}\;276(D)276(E) 284\text{(E)}\;284(E)284Related TopicsToolkit 22 — Approaches to Similar EquationsHints (8)Hint 1ab+c−bc−a=100−87=13ab+c-bc-a=100-87=13ab+c−bc−a=100−87=13Hint 2(a−c)(b−1)=13(a-c)(b-1)=13(a−c)(b−1)=13Hint 3(a−c)(b−1)=13(a−c)(b−1)131113−13−1−1−13\begin{gathered}(a-c)(b-1)=13\\[8pt]\begin{matrix}(a-c)&(b-1)\\[5pt]13&1\\[5pt]1&13\\[5pt]-13&-1\\[5pt]-1&-13\end{matrix}\end{gathered}(a−c)(b−1)=13(a−c)131−13−1(b−1)113−1−13Hint 4Case 1: a−c=13⇒a=c+13b−1=1⇒b=2ab+c=100⇒2c+26+c=100⇒3c=74⇒c=743, which is not an integer.\begin{aligned}\text{Case 1: }&a-c=13 &&\Rightarrow a=c+13\\&b-1=1 &&\Rightarrow b=2\\[7pt]&ab+c=100 &&\Rightarrow 2c+26+c=100\\&&&\Rightarrow 3c=74\\&&&\Rightarrow c=\frac{74}{3},\ \text{which is not an integer.}\end{aligned}Case 1: a−c=13b−1=1ab+c=100⇒a=c+13⇒b=2⇒2c+26+c=100⇒3c=74⇒c=374, which is not an integer.Hint 5Case 2: a−c=1⇒a=c+1b−1=13⇒b=14ab+c=100⇒14c+14+c=100⇒15c=86⇒c=8615, which is not an integer.\begin{aligned}\text{Case 2: }&a-c=1 &&\Rightarrow a=c+1\\&b-1=13 &&\Rightarrow b=14\\[7pt]&ab+c=100 &&\Rightarrow 14c+14+c=100\\&&&\Rightarrow 15c=86\\&&&\Rightarrow c=\frac{86}{15},\ \text{which is not an integer.}\end{aligned}Case 2: a−c=1b−1=13ab+c=100⇒a=c+1⇒b=14⇒14c+14+c=100⇒15c=86⇒c=1586, which is not an integer.Hint 6Case 3: a−c=−13⇒a=c−13b−1=−1⇒b=0ab+c=100⇒c=100bc+a=87⇒a=87ca+b=60⇒(100)(87)+0=60⇒8700=60×\begin{aligned}\text{Case 3: }&a-c=-13 &&\Rightarrow a=c-13\\&b-1=-1 &&\Rightarrow b=0\\[7pt]&ab+c=100 &&\Rightarrow c=100\\&bc+a=87 &&\Rightarrow a=87\\&ca+b=60 &&\Rightarrow (100)(87)+0=60\\&&&\Rightarrow 8700=60\quad\times\end{aligned}Case 3: a−c=−13b−1=−1ab+c=100bc+a=87ca+b=60⇒a=c−13⇒b=0⇒c=100⇒a=87⇒(100)(87)+0=60⇒8700=60×Hint 7Case 4: a−c=−1⇒a=c−1b−1=−13⇒b=−12ab+c=100⇒−12c+12+c=100⇒−11c=88⇒c=−8\begin{aligned}\text{Case 4: }&a-c=-1 &&\Rightarrow a=c-1\\&b-1=-13 &&\Rightarrow b=-12\\[7pt]&ab+c=100 &&\Rightarrow -12c+12+c=100\\&&&\Rightarrow -11c=88\\&&&\Rightarrow c=-8\end{aligned}Case 4: a−c=−1b−1=−13ab+c=100⇒a=c−1⇒b=−12⇒−12c+12+c=100⇒−11c=88⇒c=−8Hint 8a=c−1=−9,b=−12,c=−8These values satisfy all three equations.\begin{aligned}a&=c-1=-9,\qquad b=-12,\qquad c=-8\\[7pt]&\text{These values satisfy all three equations.}\end{aligned}a=c−1=−9,b=−12,c=−8These values satisfy all three equations.Final Answer(D) 276