A monk climbs to the top of a certain mountain with unequal speeds and random stops of random durations, he reaches the top at the sunset of the 13th day from the start. After meditating there for a week, he starts climbing down the mountain at the sunrise with unequal speeds and random stops. The speed while climbing down is obviously greater than speed climbing up. Assuming that he follows the exact same path for both journeys prove that there exists a time of day where the monk was at the same position on the path for both journeys.