1. Problem 7: Show that in any set of six classes, each meeting once a week on a weekday (Monday to Friday), there must be two classes meeting on the same day. 2. This is a classic example of the Pigeonhole Principle, which states: If $n$ items are put into $m$ containers, with $n > m$, then at least one container must contain more than one item. 3. Here, the "items" are the 6 classes, and the "containers" are the 5 weekdays. 4. Since there are 6 classes and only 5 possible days, by the Pigeonhole Principle, at least two classes must meet on the same day. 5. Problem 8: Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17$. 6. An increasing subsequence is a sequence where each term is larger than the previous one. A decreasing subsequence is the opposite. 7. To find the longest increasing subsequence (LIS), we look for the longest chain where each next number is bigger. 8. One LIS is $5, 7, 10, 15, 21$ (length 5). 9. To find the longest decreasing subsequence (LDS), we look for the longest chain where each next number is smaller. 10. One LDS is $22, 7, 2$ (length 3). 11. Thus, the maximal increasing subsequence length is 5 and the maximal decreasing subsequence length is 3.