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1. Games in a single-elimination tournament ●○○○○ 10 ptsIn a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion?
2. Matches in a knockout cup ●○○○○ 10 ptsCup football matches are played under the knockout system: the loser of each match is eliminated, and there are no rematches. 75 teams take part. How many matches must be played to decide the cup winner?
3. Distinct values of f(x) minus x ●○○○○ 10 ptsSuppose the function $f(x)$ is defined explicitly by the table $$\begin{array}{c || c | c | c | c | c} x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 0 & 0 & 1 & 3 & 6 \end{array}$$ This function is defined only for the values of $x$ listed in the table. Suppose $g(x)$ is defined as $f(x)-x$ for all numbers $x$ in the domain of $f$. How many distinct numbers are in the range of $g(x)$?
4. Two-digit primes from given digits ●○○○○ 10 ptsHow many two-digit prime numbers can be formed by choosing two different digits from the set $\{2, 7, 8, 9\}$ to be used as the tens digit and units digit?
5. Count positive factors of 1100 ●○○○○ 10 ptsHow many distinct positive factors does $1100$ have?
6. Nested divisor-count function ●○○○○ 10 ptsLet $\boxed{N}$ mean the number of positive whole number divisors of $N$. For example, $\boxed{3}=2$, because $3$ has two divisors, $1$ and $3$. Find the value of $\boxed{\boxed{11}\times\boxed{20}}$.
7. Common divisors of 48 and 156 ●○○○○ 10 ptsHow many positive divisors do 48 and 156 have in common?
8. Arrangements of the letters of TOOT ●○○○○ 10 ptsDetermine the number of distinct ways to arrange the letters of the word TOOT.
9. 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?
10. Shared sale dates in July ●○○○○ 10 ptsA bookstore has a sale on days of the month that are multiples of 5 (such as June 5, June 10, ...). A shoe store has a sale every 6 days. The shoe store has a sale on July 3. How many times in July do the two stores have sales on the same date?

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