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Olympiad workbook
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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

If a, b, c ≥ 4 are integers, not all equal, and 4 a b...

If a,b,c≥4a, b, c \geq 4 are integers, not all equal, and 4abc=(a+3)(b+3)(c+3)4 a b c=(a+3)(b+3)(c+3), then what is the value of a+b+c?a+b+c ?

Correct answer

16

Step-by-step solution

4abc=27+3(ab+bc+ca)+9(a+b+c)+abc⇒3abc=27+3(ab+bc+ca)+9(a+b+c)⇒abc=9+(ab+bc+ca)+3(a+b+c)⇒abc−(ab+bc+ca)+(a+b+c)−1=8+4(a+b+c)⇒(a−1)(b−1)(c−1)=8+4(a+b+c)\begin{aligned} & 4 a b c=27+3(a b+b c+c a)+9(a+b+c)+a b c \\ & \Rightarrow 3 a b c=27+3(a b+b c+c a)+9(a+b+c) \\ & \Rightarrow a b c=9+(a b+b c+c a)+3(a+b+c) \\ & \Rightarrow a b c-(a b+b c+c a)+(a+b+c)-1=8+4(a+b+c) \\ & \Rightarrow (a-1)(b-1)(c-1)=8+4(a+b+c) \end{aligned} Put a−1=A,b−1=B,c−1=C\mathrm{a}-1=\mathrm{A}, \mathrm{b}-1=\mathrm{B}, \mathrm{c}-1=\mathrm{C}, ⇒ABC=20+4(A+B+C)⇒A=4(5+B+C)BC−4⇒B=3,C=4,A=6\begin{array}{ll} \Rightarrow & A B C=20+4(A+B+C) \\ \Rightarrow & A=\frac{4(5+B+C)}{B C-4} \\ \Rightarrow & B=3, C=4, A=6 \end{array} or they can be interchanged ⇒(a,b,c) are anangemets of (4,5,7)⇒a+b+c=16\begin{array}{ll} \Rightarrow & (a, b, c) \text { are anangemets of }(4,5,7) \\ \Rightarrow & a+b+c=16 \end{array}
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