4. The function $y=f(x)$ is defined on the set $(0,+\infty)$ and takes positive values on it. It is known that for any points $A$ and $B$ on the graph of the function, the areas of the triangle $A O B$ and the trapezoid $A B H_{B} H_{A}$ are equal to each other $\left(H_{A}, H_{B}-\right.$ the bases of the perpendiculars dropped from points $A$ and $B$ to the x-axis; $O$ - the origin). Find all such functions. Justify your solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ec33ce2a7294e5a0d7eg-1.jpg?height=374&width=469&top_left_y=2303&top_left_x=1476)

## Interregional Olympiad for Schoolchildren Based on Departmental Educational Organizations in Mathematics
Solution. Let $M$ be the intersection point of segments $O B$ and $A H_{A}$. Since the areas of triangle $A O B$ and trapezoid $A B H_{B} H_{A}$ are equal, the areas of triangles $A M O$ and trapezoid $M B H_{B} H_{A}$ are also equal. From this, it follows that the areas of triangles $A O H_{A}$ and trapezoid $B O H_{B}$ are also equal. Let the abscissas of points $H_{A}$ and $H_{B}$ be $x$ and $t$ respectively. Then we have the equality $x \cdot f(x)=t \cdot f(t)$. For a fixed $t$, we get the conclusion: $f(x)=\frac{c}{x}, c>0$.

Answer: $f(x)=\frac{c}{x}, c>0$.