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.
2026-03-29 11:50:12.1774785012
If $ab+ac+bc=9$ and $a,b,c \geq 1,$ what is the maximum value of $a^2+b^2+c^2?$
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$$\sum_{cyc}(a-1)(b-1)\geq0$$ gives $$a+b+c\leq6$$ and by your work we are done!
The equality occurs for $(a,b,c)=(1,1,4).$