<s>[INST] If $ab+ac+bc=9$ and $a,b,c \geq 1,$ what is the maximum value of $a^2+b^2+c^2?$ If $ab+ac+bc=9$ and $a,b,c \geq 1,$ could anyone advise me how to find the maximum value of $a^2+b^2+c^2?$
I have shown that $(a+b+c)^2=a^2+b^2+c^2+18,$ so does it suffice to maximise $a+b+c ?$
Thank you.
 [/INST] $$\sum_{cyc}(a-1)(b-1)\geq0$$ gives
$$a+b+c\leq6$$ and by your work we are done!
The equality occurs for $(a,b,c)=(1,1,4).$
</s>