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Let $x,y,z\in\mathbb{R}$.Let $xy+yz+xz=1$.

Prove:$\displaystyle \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}\leq \frac{3}{2}$

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Let $a,b,c\in(0,\pi)$ be such that $x=\cot a$, $y=\cot b$, $z=\cot c$.

Using the addition formula for cotangent, one can show that

$$ \cot(a+b+c)

= \frac{\cot a\cot b\cot c – \cot a – \cot b – \cot c}

{\cot a\cot b + \cot b\cot c + \cot c\cot a – 1}

\tag{$\ast$} $$

By hypothesis, the denominator on the RHS is 0, so (see below) the LHS is $\infty$; since $a,b,c\in (0,\pi)$, this implies $a+b+c\in\{\pi,2\pi\}$.

Case $a+b+c=\pi$: Then at most one of $a,b,c$ is greater than $\frac\pi2$; wlog, $a,b\in(0,\frac\pi2]$. Since cosine is concave on that interval, we have

\begin{align*}

\frac{x}{\sqrt{x^2+1}} + \frac{y}{\sqrt{y^2+1}} + \frac{z}{\sqrt{z^2+1}}

&= \cos a + \cos b + \cos c \\

&\le 2\cos(\tfrac{a+b}{2}) + \cos c \\

&= 2\cos(\tfrac{a+b}{2}) – \cos(a+b) \\

&= 2\cos(\tfrac{a+b}{2}) – 2\cos^2(\tfrac{a+b}{2}) + 1 \\

&= \tfrac32 – 2\big(\cos(\tfrac{a+b}{2}) – \tfrac12\big)^2 \\

&\le \tfrac32

\end{align*}

with equality iff $a=b=\frac\pi3$ (whence $c=\frac\pi3$ also), that is, $x=y=z=\frac1{\sqrt3}$.

Case $a+b+c=2\pi$: Then at most one of $a,b,c$ is less than $\frac\pi2$; wlog, $a,b\in[\frac\pi2,\pi)$. Since cosine is nonpositive in that interval, we have

$$ \frac{x}{\sqrt{x^2+1}} + \frac{y}{\sqrt{y^2+1}} + \frac{z}{\sqrt{z^2+1}}

= \cos a + \cos b + \cos c

\le \cos c

\le 1

< \tfrac32

$$

Now, one annoying detail about ($\ast$): I noted that the denominator on the RHS is zero, and inferred that the RHS, and hence the LHS, is $\infty$. But what if the numerator on the RHS is also zero? Well, suppose for contradiction that both numerator and denominator are zero. Then $xyz=x+y+z$ and $xy+yz+zx=1$, and so

\begin{align*}

(t-x)(t-y)(t-z)

&= t^3 – (x+y+z)t^2 + (xy+yz+zx)t – xyz \\

&= t^3 – xyzt^2 + t – xyz \\

&= (t^2+1)(t-xyz)

\end{align*}

which is impossible because the first polynomial has three real roots (counting multiplicity) but the last has only one.

**Hint:** $$x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z)$$

Also, you will need the inequality

$$

\sqrt{AB}\le \frac12(A+B)

$$

Solution:

$$ \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}

\\= \frac{x}{\sqrt{(x+y)(x+z)}}+\frac{y}{\sqrt{(x+y)(x+z)}}

+\frac{z}{\sqrt{(x+y)(x+z)}}

\\

\le \frac12\left[

\frac x{x+y} + \frac x{x+z} +

\frac y{y+z} + \frac y{x+y} +

\frac z{z+y} + \frac z{z+x}

\right]=3/2

$$

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