<s>[INST] What is the value of $a+b+c$? 
What is the value of $a+b+c$?
if $$a^4+b^4+c^4=32$$
$$a^5+b^5+c^5=186$$
$$a^6+b^6+c^6=803$$

How to approach this kind of problem. Any help.
UPDATE: Thank you all for answers. Now I realize there is no integer solution. But is there any real number solution? I am curious to know.
 [/INST] The lowest possible values for fourth powers of integers are $$0^4 = 0 \qquad 1^4 = (-1)^4 = 1 \qquad 2^4 = (-2)^4 = 16$$
By inspection, it becomes very clear that two of $a,b,c$ must have absolute value $2$, and one must be zero, in order for your first equation to hold. 
Without loss of generality, suppose $c=0$ and $a = \pm b = \pm 2$.
Now $a^5 + b^5$ must be $2^5 + 2^5$, $-2^5 + 2^5$, or $-2^5 -2^5$. None of these is equal to $186$, so there are no solutions in the integers.
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