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3 problems match “double counting”. Clear

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1. Matches in a round-robin group ●○○○○ 10 ptsA group has four teams, and each team plays every other team in the group exactly once. How many matches are played in the group?
2. Blue segments in a two-colored grid ●●●○○ 50 ptsA $16\times16$ square grid of dots is colored red and blue. Two dots that are adjacent in the grid (horizontally or vertically) are joined by a line segment. A segment joining two dots of the same color has that color, and a segment joining dots of different colors is yellow. There are 133 red dots in total. Exactly 2 of the 4 corner dots are red, and 32 other red dots lie on the boundary of the grid (not corners). There are 196 yellow segments in total. How many blue segments are there?
3. Minimum number of 7-subsets covering all triples ●●●●○ 100 ptsLet $S=\{1,2,\ldots,15\}$. Choose $n$ subsets $A_1,A_2,\ldots,A_n$ of $S$ such that: (i) $|A_i|=7$ for each $i$; (ii) $|A_i\cap A_j|\leq 3$ for all $1\leq i<j\leq n$; (iii) for every 3-element subset $M$ of $S$, there is some $k$ with $M\subset A_k$. Find the minimum possible value of $n$.