Problem 1. A student started working on an exam between 9 and 10 o'clock and finished between 13 and 14 o'clock. Find the exact time spent by the applicant on completing the task, given that at the beginning and the end of the work, the hour and minute hands, having swapped places, occupied the same positions on the clock face.

Answer: $60 / 13$ hours

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# Solution.

From the initial and final positions of the hour hand, it is clear that the student started writing before 9:30 and finished after 1:30 PM. Therefore, the exact time \( x \), spent on the work, satisfies the inequalities \( 4 < x < 5 \). If after completing the work, the minute hand travels the distance the hour hand had traveled from the initial position of the minute hand on the clock face, this will add \( x/12 \) hours to \( x \), since the minute hand moves 12 times faster than the hour hand. As a result, from the start of the exam, \( x + x / 12 \) hours will pass, and this number must be an integer, as the minute hand will complete a whole number of revolutions, and it must be greater than \( x \) and the closest to \( x \). Therefore, \( x + x / 12 = 5 \), \( x = 60 / 13 \) hours.