Find the largest positive integer $n$ for which the inequality

$$
\frac{a+b+c}{a b c+1}+\sqrt[n]{a b c} \leq \frac{5}{2}
$$

holds for all $a, b, c \in[0,1]$. Here $\sqrt[1]{a b c}=a b c$.
Let $n_{\max }$ be the sought largest value of $n$, and let $E_{a, b, c}(n)=\frac{a+b+c}{a b c+1}+\sqrt[n]{a b c}$. Then $E_{a, b, c}(m)-E_{a, b, c}(n)=\sqrt[m]{a b c}-\sqrt[n]{a b c}$ and since $a b c \leq 1$ we clearly have $E_{a, b, c}(m) \geq E_{a, b, c}(n)$ for $m \geq n$. So if $E_{a, b, c}(n) \geq \frac{5}{2}$ for some choice of $a, b, c \in[0,1]$, it must be $n_{\max } \leq n$. We use this remark to determine the upper bound $n_{\max } \leq 3$ by plugging some particular values of $a, b, c$ into the given inequality as follows:

$$
\text { For }(a, b, c)=(1,1, c), c \in[0,1] \text {, inequality (1) implies } \frac{c+2}{c+1}+\sqrt[n]{c} \leq \frac{5}{2} \Leftrightarrow \frac{1}{c+1}+\sqrt[n]{c} \leq
$$

$\frac{3}{2}$. Obviously, every $x \in[0 ; 1]$ is written as $\sqrt[n]{c}$ for some $c \in[0 ; 1]$. So the last inequality is equivalent to:

$$
\begin{aligned}
& \frac{1}{x^{n}+1}+x \leq \frac{3}{2} \Leftrightarrow 2+2 x^{n+1}+2 x \leq 3 x^{n}+3 \Leftrightarrow 3 x^{n}+1 \geq 2 x^{n+1}+2 x \\
\Leftrightarrow & 2 x^{n}(1-x)+(1-x)+(x-1)\left(x^{n-1}+\cdots+x\right) \geq 0 \\
\Leftrightarrow & (1-x)\left[2 x^{n}+1-\left(x^{n-1}+x^{n-2}+\ldots+x\right)\right] \geq 0, \forall x \in[0,1]
\end{aligned}
$$

For $n=4$, the left hand side of the above becomes $(1-x)\left(2 x^{4}+1-x^{3}-x^{2}-x\right)=$ $(1-x)(x-1)\left(2 x^{3}+x^{2}-1\right)=-(1-x)^{2}\left(2 x^{3}+x^{2}-1\right)$ which for $x=0.9$ is negative. Thus, $n_{\max } \leq 3$ as claimed.

Now, we shall prove that for $n=3$ inequality (1) holds for all $a, b, c \in[0,1]$, and this would mean $n_{\max }=3$. We shall use the following Lemma:

Lemma. For all $a, b, c \in[0 ; 1]: a+b+c \leq a b c+2$.
Proof of the Lemma: The required result comes by adding the following two inequalities side by side

$$
\begin{aligned}
& 0 \leq(a-1)(b-1) \Leftrightarrow a+b \leq a b+1 \Leftrightarrow a+b-a b \leq 1 \\
& 0 \leq(a b-1)(c-1) \Leftrightarrow a b+c \leq a b c+1
\end{aligned}
$$

Because of the Lemma, our inequality (1) for $n=3$ will be proved if the following weaker inequality is proved for all $a, b, c \in[0,1]$ :

$$
\frac{a b c+2}{a b c+1}+\sqrt[3]{a b c} \leq \frac{5}{2} \Leftrightarrow \frac{1}{a b c+1}+\sqrt[3]{a b c} \leq \frac{3}{2}
$$

Denoting $\sqrt[3]{a b c}=y \in[0 ; 1]$, this inequality becomes:

$$
\begin{aligned}
& \frac{1}{y^{3}+1}+y \leq \frac{3}{2} \Leftrightarrow 2+2 y^{4}+2 y \leq 3 y^{3}+3 \Leftrightarrow-2 y^{4}+3 y^{3}-2 y+1 \geq 0 \\
\Leftrightarrow & 2 y^{3}(1-y)+(y-1) y(y+1)+(1-y) \geq 0 \Leftrightarrow(1-y)\left(2 y^{3}+1-y^{2}-y\right) \geq 0
\end{aligned}
$$

The last inequality is obvious because $1-y \geq 0$ and $2 y^{3}+1-y^{2}-y=y^{3}+(y-1)^{2}(y+1) \geq 0$.

## Geometry

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