REMT 2016
MATHEMATICAL
LOGISTICS
ANSWERS
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LAST NAME
FIRST NAME
GRADE
2016H2
1.
Compute the unique positive integer that, when squared, is equal to six more than five
times itself.
2.
Solve for 𝑥 in the equation √√2 − 8 = −1 .
3.
Define a set of positive integers to be balanced if the set is not empty and the number of
even integers in the set is equal to the number of odd integers in the set. How many
subsets of the set of the first 10 positive integers are balanced?
4.
Alice and Bob are playing a game in which Alice has a 1/3 probability of winning, a 1/2
probability of tying, and a 1/6 probability of losing. Given that Alice and Bob played a
game which did not end in a tie, compute the probability that Alice won.
5.
If f and g are functions such that f(x) = 3x + 2 and f(g(x)) = 5x + 4, write a formula for
g(x).
3
𝑥
2016H3
6.
For what value of 𝑎 is 𝑥 − 2 a factor of 𝑥 4 + 𝑎𝑥 3 − 4𝑥 2 + 2𝑥 − 1 ?
7.
Let 𝑓 be a function of a real variable with the properties that 𝑓(𝑥) + 𝑓(1 − 𝑥) = 11 and
𝑓(1 + 𝑥) = 3 + 𝑓(𝑥) for all real 𝑥. What is the value of 𝑓(𝑥) + 𝑓(−𝑥) ?
8.
Write an equation for the slant asymptote to the graph of 𝑦 =
9.
Find the remainder when 𝑓(𝑥) = 𝑥 81 + 𝑥 32 + 𝑥 6 + 3𝑥 2 + 1 is divided by 𝑥 2 − 1 .
10.
Find the number of different routes from point A
to point B always heading north or east.
2𝑥 2 −3𝑥+4
𝑥−4
.
B
A
2016H4
11.
Let 𝑥 = √16 + √16 + √16 + ⋯ . What is the value of x ?
12.
An infinite geometric sequence has a first term of 12, and all terms in the sequence sum
to 9. Compute the common ratio between consecutive terms of the geometric sequence.
13.
An ant begins at a vertex of a cube. On each move, it travels along an edge to a randomly
selected adjacent vertex. Find the probability that it is back at its starting position after 4
moves.
14.
Find the remainder when 220 + 330 + 440 + 550 + 660 is divided by 9.
15.
It takes three lumberjacks three minutes to saw three logs into three pieces each. How
many minutes does it take six lumberjacks to saw six logs into six pieces each?