1. At what moment after noon do the hour and minute hands form a right angle for the first time?
Answer: $180 / 11$ minutes $=164 / 11$ minutes.

## Solution:

The hour hand makes 1 revolution in 12 hours, while the minute hand makes 12 revolutions in the same time. Therefore, the minute hand moves 12 times faster. We will measure everything in minutes, so a right angle corresponds to 15 minutes. Let the hour and minute hands form a right angle at time $\mathrm{t}$ - during this time, the minute hand will travel a "distance" (angle) of $t$, and the hour hand will travel 12 times less. Therefore, the condition of the problem can be written as follows:

$\mathrm{t}-\mathrm{t} / 12=15$, then $\mathrm{t}=180 / 11$ minutes.

## Criteria:

Correct exact answer (without statements like "approximately 16 minutes") - 2 points.

Correctly written equation for the angle (from the speed ratio or similar considerations) - another 3 points.

Fully correct solution - 7 points.