<s>[INST] Find the largest integer that satisfies this system $a,b,c,d$ are real numbers. Find the largest integer value of $d$ such that
$$a+b+c=d$$
$$a^2+b^2+c^2=d+1$$
The two equations seem to be of a plane and a sphere. But how do I proceed from here? Thanks.
 [/INST] The following may be useful. By the Cauchy-Schwarz Inequality, we have $(a+b+c)^2 \le (a^2+b^2+c^2)(3)$. In the notation of the linked article, we use $y_i=1$ for $i=1,2,3$. 
Thus if $a+b+c=d$ and $a^2+b^2+c^2=d+1$, we have $d^2 \le 3(d+1)$. If $d$ is to be an integer, this forces $d\le 3$. 
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