Let's determine the minimum of the expression $z=x+y+\sqrt{(a-x)^{2}+(b-y)^{2}}$, where $a$ and $b$ are real numbers, and $x$ and $y$ are arbitrary non-negative real numbers.
# I. Solution.

For the (possibly degenerate) triangle shown in Figure 1,

$$
O P+P A \geqq O A
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_f281ec18176883b63661g-1.jpg?height=363&width=390&top_left_y=393&top_left_x=846)

Figure 1

Expressing the distances involved, (1) becomes:

$$
\sqrt{x^{2}+y^{2}}+\sqrt{(a-x)^{2}+(b-y)^{2}} \geqq \sqrt{a^{2}+b^{2}}
$$

Since $x$ and $y$ are non-negative numbers, we have

$$
x+y=\sqrt{x^{2}+2 x y+y^{2}} \geqq \sqrt{x^{2}+y^{2}}
$$

Thus, based on (2) and (3),

$$
z=x+y+\sqrt{(a-x)^{2}+(b-y)^{2}} \geqq \sqrt{x^{2}+y^{2}}+\sqrt{(a-x)^{2}+(b-x)^{2}} \geqq \sqrt{a^{2}+b^{2}}
$$

Therefore, the minimum value of $z$ is $\sqrt{a^{2}+b^{2}}$, which is achieved when $x=y=0$ for non-negative $x$ and $y$.

Németh Sándor (Györ, Révai M. Gymnasium IV. grade)

## II. Solution.

![](https://cdn.mathpix.com/cropped/2024_05_02_f281ec18176883b63661g-1.jpg?height=377&width=548&top_left_y=1470&top_left_x=775)

Figure 2

Using the notation in Figure 2,

$$
\begin{array}{r}
z=A C+C D+D B= \\
=x+y+\sqrt{(a-x)^{2}+(b-y)^{2}}
\end{array}
$$

Thus, $z$ is the length of the broken line $A C D B$. This is clearly minimized when the line is a straight segment, i.e.,

$$
z=A B=\sqrt{a^{2}+b^{2}}
$$

This occurs precisely when $x=0$ and $y=0$ for non-negative $x$ and $y$.

Remark: In both solutions, we heavily utilized the fact that $x$ and $y$ are non-negative, as in the second solution, $x$ and $y$ appeared as lengths of segments. The problem can be generalized from the plane to space, or even to $n$-dimensional space, making the following statement valid:

If $a_{1}, a_{2}, \cdots, a_{n}$ are given real numbers, and $x_{1}, x_{2}, \cdots, x_{n}$ are arbitrary non-negative numbers, then the expression

$$
z=x_{1}+x_{2}+\cdots+x_{n}+\sqrt{\left(a_{1}-x_{1}\right)^{2}+\left(a_{2}-x_{2}\right)^{2}+\cdots+\left(a_{n}-x_{n}\right)^{2}}
$$

has a minimum value of $\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}$.