<s>[INST] Maximum value with inequality This is a problem that my friend and I are working on for olympiad training.

Let $a, b, c$ be real numbers in the interval $[0,1]$ that satisfy $ab+c \leq 1$. What is the maximum value of $a+b+c?$

I'm guessing the maximum is at $a,b=1$ and $c=0$, where we have $a+b+c=2$. As for proving this, I'm not sure how to proceed. Maybe AM-GM? Thanks for the help.
 [/INST] 1)$ab+c\le 1;$ $a, b, c \in [0,1];$
$d:=a+b+c$
The $2$ expressions are symmetric in $a$ and $b$. It follows that $a=b$ for $d_{max}$.
2)We want to find the maximum of
$d=2a+c,$ with $a^2+c \le 1$.
3)$a^2+d-2a \le 1;$
$(a-1)^2-1 +d \le 1;$
$d \le 2 - (a-1)^2.$
4)Maximal value for $d_{max}=2$ at $a=1;$
Hence $a=b=1$,and $c=0$.
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