if real numbers a,b,c,d,e satisfy

a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3

what is the value of a^2+b^2+c^2+d^2+e^2 ?

The answer is 10.

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Let $a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3=k$ .

Now $a+1=k$ .

Rearranging we get $a=k-1$ .

Similarly, $b=k-2$ , $c=k-3$ , $d=k-4$ , $e=k-5$ .

Also $a+b+c+d+e+3=k$ .

Substituting the values in we get $k=3$ .

Now $a=2$ , $b=1$ , $c=0$ , $d=-1$ , $e=-2$ .

Therefore $\boxed{{a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}+{e}^{2}=10}$ .