# Proving $\sum\limits_{\mathrm{cyc}} \sqrt{7x+7y-8}\ge3\sqrt{6}$ when $x, y, z\in [0, 5/2]$ with $xy+yz+zx+xyz=4$

Problem. Let $$x,y,z$$ in $$\left[0;\frac{5}{2}\right]$$ such that $$xy+yz+zx+xyz=4$$. Prove that: $$\sqrt{7x+7y-8}+\sqrt{7y+7z-8}+\sqrt{7z+7x-8}\ge3\sqrt{6}.$$

The inequality was posted in AOPS.

I've tried to use substitution $$x=\frac{2a}{b+c},y=\frac{2b}{a+c},z=\frac{2c}{b+a}$$ and we'll prove $$\sum_{cyc}\sqrt{7\left(\frac{a}{b+c}+\frac{b}{a+c}\right)-4}\ge3\sqrt{3}.$$

But I don't know how to use the hypothesis $$x,y,z$$ in $$\left[0;\frac{5}{2}\right].$$

I need some hints to deal with this strange condition. Thank you for your help.

Some thoughs.

Fact 1. Let $$a, b, c \ge 0$$ with $$a^2b^2+b^2c^2+c^2a^2 + \frac{11}{32}a^2b^2c^2 \ge \frac{729}{4}$$. Then $$a + b + c \ge 3\sqrt{6}$$.
(Note: It is verified by Mathematica. A human verifiable proof is needed.)

Now, let $$a := \sqrt{7x + 7y - 8}, \quad b := \sqrt{7y + 7z - 8}, \quad c := \sqrt{7z + 7x - 8}.$$ We need to prove that $$a + b + c \ge 3\sqrt{6}$$. By Fact 1, it suffices to prove that $$a^2b^2+b^2c^2+c^2a^2 + \frac{11}{32}a^2b^2c^2 \ge \frac{729}{4}$$ or $$\sum_{\mathrm{cyc}} (7x+7y-8)(7y+7z-8) + \frac{11}{32}(7x+7y-8)(7y+7z-8)(7z+7x-8) \ge \frac{729}{4}. \tag{1}$$

If $$xyz = 0$$, it is easy.

In the following, assume that $$xyz \ne 0$$.

Let $$u := \frac{\frac52 - x}{x}, \quad v := \frac{\frac52 - y}{y}, \quad w := \frac{\frac52 - x}{x}.$$ We have $$u, v, w \ge 0$$ and $$x = \frac52\cdot \frac{1}{1 + u}, \quad y = \frac52\cdot \frac{1}{1 + v}, \quad z = \frac52\cdot \frac{1}{1 + w}.$$

Let $$p = u + v + w, q = uv + vw + wu, r = uvw$$.

(1) is equivalent to \begin{align*} &9206\,{p}^{2}-3949\,pq-3280\,pr-3600\,{q}^{2}-896\,qr-1216\,{r}^{2}\\ &\quad + 50328\,p-4618\,q-23059\,r+64152\\ &\ge 0. \tag{2} \end{align*} The condition $$xy + yz + zx + xyz = 4$$ is written as $$18p - 32q - 32r + 243 = 0. \tag{3}$$

Also, we have $$0 \le (u-v)^2(v-w)^2(w-u)^2 = -4\,{p}^{3}r+{p}^{2}{q}^{2}+18\,pqr-4\,{q}^{3}-27\,{r}^{2}. \tag{4}$$

Thus, it suffices to prove that (2) is true for all $$p, q, r \ge 0$$ satisfying (3) and (4). The rest is smooth.