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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of triples of real numbers (a, b, c) such that a^...

Find the number of triples of real numbers (a,b,c)(a, b, c) such that a20+b20+c20=a24+b24+c24=1a^{20}+b^{20}+c^{20}=a^{24}+b^{24}+c^{24}=1.

Correct answer

6

Step-by-step solution

a20+b20+c20=a24+b24+c24=1a^{20}+b^{20}+c^{20}=a^{24}+b^{24}+c^{24}=1 If both equation have to satisfy ⇒\Rightarrow \quad Either aa or bb or c=±1c= \pm 1 And other have to be 0 . So, if a=1a=1 ⇒b=c=0\Rightarrow \quad b=c=0 If a=−1a=-1 ⇒b=c=0\Rightarrow \quad b=c=0 Similarly, for bb and cc if b=±1b= \pm 1 ⇒a=c=0\Rightarrow \quad a=c=0 And if c=±1c= \pm 1 ⇒a=b=0\Rightarrow \quad a=b=0 Total triples =6=6.
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