CO 380 Spring 2020
1. The sequence of positive integers a1, a2, a3, . . . satisfies a1 = c, a2 = 10 and an+2 = 5an+1 +10an
for each positive integer n ≥ 1. Determine all possible values of c for which one of the terms in
the sequence equals 2000.
2. Determine, with justification, the number of ways of tiling a 6 × 15 rectangle with 1 × 6 tiles.
3. At Neumann H.S., there are 234 students. Of these, 156 students are studying German and 165
studying Polish. Suppose that m is the mininum possible number of students studying both
German and Polish. Suppose that M is the maximum possible number of students studying
both German and Polish. Determine the values of m and M.
4. The line y = mx, with m 6= 0 and m 6= −1 is reflected in the line x + y = 1. Determine the
slope and the y-intercept of the resulting line, in terms of m.
5. Determine all real x that satisfy log2x
(48√3
3) = log3x
(162√3
2).
6. Sally the red-footed tortoise is travelling at 6 m/h on a long straight road parallel to a turtle
path. Every 10 minutes, Sally is passed by a turtle moving in the same direction as she is.
These turtles leave from a turtle farm behind her every 3 minutes and all move at the same
constant speed along the turtle path. Measured in m/h, what is the constant speed of the
turtles?
7. S = 4 − 5 + 1 − 7 + 2 − 6 + 3 and S = 1 − 2 + 3 − 4 + 5 − 6 + 7 are two examples of the
alternating sum of the positive integers from 1 to 7, inclusive. What is the average value of S
over all such alternating sums of the positive integers from 1 to 7, inclusive.
(Each alternating sum contains each of the integers 1 through 7 exactly once, and the operations
alternate with subtraction first, addition second, subtraction third, and so on.)
8. For any real number x, bxc denotes the largest integer less than or equal to x.
For each positive integer n, determine all possible integer values of the expression x
2 − 3bxc,
where x is a real number with bxc = n.

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