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1. Blue points on a line ●●○○○ 20 ptsOn a line, one hundred points are marked, each painted green, blue, or red. Between any two red points there is a blue point, and between any two blue points there is a green point. There are no fewer red points than blue points, and no fewer blue points than green points. How many points are painted blue?
2. Range of a five-integer set ●●○○○ 20 ptsA set of five positive integers has mean 5, median 5, and 8 as its only mode. What is the difference between the largest and smallest integers in the set?
3. Distinct candy flavors by percent red ●●○○○ 20 ptsFour red candies and three green candies can be combined to make many different flavors. Two flavors are different if their percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green, and 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is made using some or all of the seven candies (at least one candy), how many different flavors are possible?
4. Largest number of jogs ●●○○○ 20 ptsBill can buy jags, jigs and jogs for $1, $2 and $7 each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $50?
5. Largest integer in a five-number list ●●○○○ 20 ptsA list of five positive integers has all of the following properties: - The only integer in the list that occurs more than once is $8$. - Its median is $9$. - Its average (mean) is $10$. What is the largest possible integer that could appear in the list?
6. Smallest maximum rounding error ●●○○○ 20 ptsThe sum of the following seven numbers is exactly 19: $a_1 = 2.56,$ $a_2 = 2.61,$ $a_3 = 2.65,$ $a_4 = 2.71,$ $a_5 = 2.79,$ $a_6 = 2.82,$ $a_7 = 2.86.$ Each $a_i$ is approximated by an integer $A_i,$ for $1 \le i \le 7,$ such that the sum of the $A_i$'s is also $19.$ Let $M$ be the maximum of the seven errors $|A_i - a_i|.$ What is the smallest possible value of $M$? Give your answer as a decimal.
7. Primes from up to two digits ●●○○○ 20 ptsPick one or two different digits from the set $\{1, 3, 6, 7\}$ and arrange them to form a number. How many prime numbers can be created in this manner?
8. Almost palindromic letter arrangements ●●●○○ 50 ptsCall a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost palindrome.
9. 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?
10. Counting topologies on three points ●●●○○ 50 ptsLet $X$ be a set with $3$ elements. How many topologies can be defined on $X$? Equivalently, how many subsets of the power set $\mathcal{P}(X)$ are topologies on $X$?

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