What is the smallest size plane mirror a 184-cm {\rm cm}-tall person should buy to see himself if he mounts the mirror on a vertical wall?

To determine the size of the plane mirror that a 184-cm tall person should buy to see themselves when mounted on a vertical wall, we need to consider the minimum height required for the person to see their entire reflection.

In order to see the full reflection, the person's eye level should be positioned at the midline of the mirror. This means that the vertical distance from the person's eye level to the bottom of the mirror should be equal to the distance from the person's eye level to the top of the mirror.

Let's call the height of the mirror "h". Therefore, the vertical distance from the person's eye level to the center of the mirror is h/2.

The person's eye level will be located at 184 cm / 2 = 92 cm above the ground.

To calculate the required height of the mirror, we need to add the vertical distance from the eye level to the center of the mirror, which is h/2.

So, the total height of the mirror will be 92 cm + h/2.

Since the bottom of the mirror is the same distance from the person's eye level as the top of the mirror, the bottom of the mirror is also located at 92 cm below the person's eye level.

To sum up, the total height of the mirror is h, and the bottom of the mirror is positioned 92 cm below the person's eye level.

Therefore, the smallest size plane mirror a 184-cm tall person should buy to see himself, when mounted on a vertical wall, is a mirror with a height of 92 cm + h/2.

This calculation assumes that the person's eye level is located at the center of their height. If you prefer a different position for the eye level, you can adjust the calculations accordingly.