MHB What is the Minimum Value of Sum of Squares for Real Numbers in this POTW?

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The problem involves finding the minimum value of the sum of squares of four real numbers, a, b, c, and d, given three equations that relate their sums. The equations are (a+b)(c+d)=13, (a+c)(b+d)=24, and (a+d)(b+c)=25. A solution was provided by user castor28, who successfully calculated the minimum value. The discussion emphasizes the importance of understanding the relationships between the variables to solve the problem effectively. The thread highlights the collaborative nature of problem-solving in mathematics.
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Here is this week's POTW:

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Let $a,\,b,\,c$ and $d$ be real numbers that satisfy

$(a+b)(c+d)=13,\\(a+c)(b+d)=24,\\(a+d)(b+c)=25.$

Find the smallest possible vale of $a^2+b^2+c^2+d^2$.

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Congratulations to the castor28 for his correct solution, which you can find below: (Smile)
We note that the given equations imply that $(a+b+c+d)^2 = a^2 + b^2 + c^2 + d^2 + 13 + 24 + 25$.

If we write $t=a+b+c+d$, $x=a+b$, $y=a+c$, and $z=a+d$, the equations can be written as:
$$\begin{align*}
x(t-x) &= 13\qquad[1]\\
y(t-y) &= 24\qquad[2]\\
z(t-z) &= 25\qquad[3]
\end{align*}
$$

As these quadratic equations must have real solutions, their discriminants must be non-negative. The strongest condition comes from the third equation and gives $t^2\ge100$. As changing the sign of $t$ merely changes the signs of all the variables, we may assume that $t\ge10$.

It remains to show that this value can be effectively attained. We may compute a solution explicitly. Taking $t = 10$ and choosing $y_1=6$ and $z_1=5$ as roots of the least two equations, we get the system:
$$\begin{align*}
a+c &=6\\
b+d &= 4\\
a+d&=5\\
b+c&=5
\end{align*}
$$

and the general solution of that system is $a=5-d$, $b=4-d$, $c=d+1$. Substituting in the first given equation and solving for $d$, we get $d=2\pm\sqrt3$. This gives one possible solution:
$$\begin{align*}
a &= 3- \sqrt3\\
b &= 2-\sqrt3\\
c &= 3+\sqrt3\\
d &= 2+\sqrt3
\end{align*}
$$
 
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