The answer is 3.

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Let f(x) = x^x. We can see that the graph of x^x is always positive having a minima at x =1/e. Basically it is a convex graph. Using Jensen's inequality for a convex graph , we have (f(a) + f(b) + f(c))/3 >= f( (a+b+c)/3). Equality holds when a=b=c. Thus a^a + b^b + c^c} >= 3f(1) as a+b+c=3. f(1)=1^1=1. Hence required answer is 3.