Finding the maximum value of $ab+ac+ad+bc+bd+3cd$

If $a,b,c,d>0$ satisfy the condition ${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 }=1$, find the maximum value of $ab+ac+ad+bc+bd+3cd$.

Thank you.

• You could try langrange multipliers – XPenguen Dec 22 '15 at 15:08
• Easily, but I'm in search of an elementary solution suitable for contest math. – Swapnil Das Dec 22 '15 at 15:21
• Use Couchy inequality in first equation. Then consider the second. – openspace Dec 22 '15 at 15:26

Hint: consider that the eigenvalues of the symmetric matrix $$M = \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 3 \\ 1 & 1 & 3 & 0 \end{pmatrix}$$ are $-3,-1,2+\sqrt{5}$ and $2-\sqrt{5}$. In particular, $\left(\frac{\sqrt{5}-1}{2},\frac{\sqrt{5}-1}{2},1,1\right)^T$ in the only eigenvector of $M$ with positive coordinates, and it is associated with the largest eigenvalue $2+\sqrt{5}$. It follows that:
$$\max_{\substack{a^2+b^2+c^2+d^2=1 \\ a,b,c,d>0}} \frac{1}{2}(a\, b\, c\, d)\, M\, (a\, b\, c\, d)^T = \frac{2+\sqrt{5}}{2} = \color{red}{1+\frac{\sqrt{5}}{2}}.$$
Such a maximum is attained by $(a,b,c,d)=(x,x,y,y)$ with $x=\frac{1}{2}\sqrt{1-\frac{1}{\sqrt{5}}}$ and $y=\frac{1}{2}\sqrt{1+\frac{1}{\sqrt{5}}}$.
Let $ab+ac+ad+bc+bd+3cd=k$. Hence, $k>0$ and $ab+ac+ad+bc+bd+3cd=k(a^2+b^2+c^2+d^2)$ or $ka^2-(b+c+d)a+k(b^2+c^2+d^2)-bc-bd-3cd=0$. Hence, $(b+c+d)^2-4k(k(b^2+c^2+d^2)-bc-bd-3cd)\geq0$ or $(4k^2-1)b^2-2(2k+1)(c+d)b+(4k^2-1)(c^2+d^2)-2(6k+1)cd\leq0$. If $0<k\leq\frac{1}{2}$ so the last inequality is obviously true. Let $k>\frac{1}{2}$. Hence, $(2k+1)^2(c+d)^2-(4k^2-1)\left((4k^2-1)(c^2+d^2)-2(6k+1)cd\right)\geq0$ or $(2k+1)(c+d)^2-(2k-1)\left((4k^2-1)(c^2+d^2)-2(6k+1)cd\right)\geq0$ or $(2k^2-k-1)c^2-(6k-1)cd+(2k^2-k-1)d^2\leq0$. If $\frac{1}{2}<k\leq1$ so the last inequality is obviously true. Let $k>1$. Hence, $(6k-1)^2-4(2k^2-k-1)^2\geq0$, which gives $k\leq1+\frac{\sqrt5}{2}$. Easy to see that for $k=1+\frac{\sqrt5}{2}$ the equality indeed occurs. Id est, the answer is $1+\frac{\sqrt5}{2}$.