We have
a+b+c=3\cdots(1)
a^2+b^2+c^2=3\cdots(2)
Squaring (1) we get
a^2+b^2 + c^2 + 2ab + 2bc + 2ca = 9
Putting the value of a^2+b^2+c^2 from (2) we get
ab+bc+ca = 3
subracting above from (2) we get
a^2+b^2+c^2 - ab - bc - ca = 0
or 2a^2+2b^2+2c^2 - 2ab - 2bc - 2ca = 0
or (a^2-2ab + b^2) + (b^2-2bc+c^2) + (c^-2ca +a^2) = 0
or (a-b)^2 + (b-c)^2 + (c-a)^2= 0
above is possible only if a=b= c
putting in (1) we get a = b = c = 1
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