For positive a,b,c,d, if $a^2+b^2+c^2+d^2+abcd=5$, show $a+b+c+d\leq 4$.

Question

One can use Lagrange multiplier, but I am looking for a more elementary proof.

I try to find the maximum of $a+b+c+d$ and follow the standard approach.

Construct $a+b+c+d+\lambda (a^2+b^2+c^2+d^2+abcd-5)$.

One can obtain $1+2a\lambda +bcd =0$ and so on, and eventually have $a=b=c=d$.

Then you are done.

Answer 1

Let $a+b+c+d>4$, $a=kx$, $b=ky$, $c=kz$ and $d=kt$ such that $k>0$ and $$x+y+z+t=4.$$ Thus, $$k(x+y+z+t)>4,$$ which gives $$k>1.$$ But, $$5=a^2+b^2+c^2+d^2 + abcd=k^2(x^2+y^2+z^2+t^2)+k^4xyzt>x^2+y^2+z^2+t^2+xyzt,$$ which is a contradiction because we'll prove now that $$x^2+y^2+z^2+t^2+xyzt\geq5$$ or $$16(x^2+y^2+z^2+t^2)(x+y+z+t)^2+256xyzt\geq5(x+y+z+t)^4.$$ Indeed, let $x=\min\{x,y,z,t\}$, $y=x+u$, $z=x+v$ and $t=x+w$.

Thus, $$16(x^2+y^2+z^2+t^2)(x+y+z+t)^2+256xyzt-5(x+y+z+t)^4=$$ $$=32\sum_{cyc}(3u^2-2uv)x^2+16\sum_{cyc}(5u^3-u^2v-u^2w)x+$$ $$+(u+v+w)^2\sum_{cyc}(11u^2-10v^2)\geq0$$ and we are done!

Answer 2

Alternative solution:

Fact 1: For $a, b, c, d \ge 0$, it holds that $$a^2 + b^2 + c^2 + d^2 + abcd - 5 - 3(a+b+c+d - 4) \ge 0.\tag{1}$$

From Fact 1, the desired result follows.

Edit 2022/02/14: A nice proof of Fact 1 is given by mudok@AoPS.
See: #4 in https://artofproblemsolving.com/community/c6t243f6h2780412_hard_inequality

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Proof of Fact 1:

From Vasc's Equal Variable Theorem [1, Corollary 1.9], we only need to prove the case when $a = b = c$.

It suffices to prove that $d^2+(a^3-3)d+3a^2-9a+7 \ge 0$.

Since $3a^2 - 9a + 7 > 0$, by AM-GM, we have $d^2+(a^3-3)d+3a^2-9a+7 \ge (a^3-3)d + 2d\sqrt{3a^2-9a+7}$.
It suffices to prove that $a^3 - 3 + 2\sqrt{3a^2-9a+7} \ge 0$.

If $a > \frac{3}{2}$, then $a^3 > 3$ and the inequality is true.

If $a \le \frac{3}{2}$, then it suffices to prove that $4(3a^2-9a+7) - (a^3 - 3)^2 \ge 0$ or $(-a^4 - 2a^3 - 3a^2 + 2a + 19)(a-1)^2 \ge 0$.

We have $-a^4 - 2a^3 - 3a^2 + 2a + 19 \ge - (3/2)^4 - 2\cdot (3/2)^3 - 3\cdot (3/2)^2 + 19 > 0$.

We are done.

Reference

[1] Vasile Cirtoaje, “The Equal Variable Method”, J. Inequal. Pure and Appl. Math., 8(1), 2007. https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf