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14 problems match “combinatorics”. Clear

Matches in a knockout tournament ●●○○○ 20 ptsThere are 100 nations taking part in an elimination tournament, in which a team is eliminated as soon as it loses a match. How many matches must be played in total to determine the overall winner?
Possible resistances from four resistors ●●●○○ 50 ptsRebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistances $a$ and $b$ ohms, and combines them into one by one of the following methods: (i) connect them in series, producing a resistor of $a+b$ ohms; (ii) connect them in parallel, producing a resistor of $\frac{ab}{a+b}$ ohms; (iii) short-circuit one of the two resistors, producing a resistor of either $a$ or $b$ ohms. Suppose that after three minutes, Rebecca has a single resistor with resistance $R$ ohms. How many possible values are there for $R$ ?
Maximum number of quadratic equations on a board ●●●○○ 50 ptsWhat is the maximum number of different monic quadratic equations that can be written on a board, given that any two of them have a common root, but no four of them have a root common to all?
Tournament with two dropouts ●●●○○ 50 ptsIn a tournament, each participant was supposed to play exactly one game with each of the other participants. Two participants dropped out during the tournament, each having played only 4 games. In the end, the total number of games played was 62. How many participants were there in total?
Counting nine-term sequences with ratio constraints ●●●○○ 50 ptsThe sequence $\{a_n\}$ has nine terms, $a_1=a_9=1$ , and for each $i\in\{1,2,\ldots,8\}$ we have $\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}$ . How many such sequences are there?
Minimum operations to clear 2021 piles ●●●○○ 50 ptsThere are 2021 piles of stones, with the number of stones in each pile being $1, 2, \ldots, 2021$ respectively. In one operation you select any number of piles and remove the same number of stones from each selected pile. What is the minimum number of operations required to remove all the stones?
Distinct values from parenthesised divisions ●●●○○ 50 ptsHow many different values can be obtained by placing parentheses in all possible ways in the expression $2:3:5:7:11:13:17:19:23:29$ , where $:$ denotes division?
Counting alternating-parity increasing sequences ●●●○○ 50 ptsA finite strictly increasing sequence of integers is called alternating if its first term is odd, its second term is even, its third term is odd, and so on, alternating in parity. The empty sequence also counts as alternating. Let $A(n)$ be the number of alternating sequences whose terms are all taken from $\{1,2,\ldots,n\}$ , so $A(1)=2$ and $A(2)=3$ . Find $A(20)$ .
Matches played in a football tournament ●●●●○ 100 ptsA football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. Each pair of teams plays at most one match, and teams from the same city do not play each other. After several days, every team except Team A from City A has played a different number of matches. How many matches has Team B from City A played?
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 $n$ .