Question 1
Kymbrea's comic book collection currently has 
comic books in it, and she is adding to her collection at the rate of 
comic books per month. LaShawn's collection currently has 
comic books in it, and he is adding to his collection at the rate of 
comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
Solution
Question 2
Real numbers 
, 
, and 
satify the inequalities 
, 
, and 
. Which of the following numbers is necessarily positive?
Solution
Question 3
Supposed that and are nonzero real numbers such that 
. What is the value of 
?
Solution
Question 4
Samia set off on her bicycle to visit her friend, traveling at an average speed of 
kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 
kilometers per hour. In all it took her 
minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Solution
Question 5
The data set 
has median 
, first quartile 
, and third quartile 
. An outlier in a data set is a value that is more than 
times the interquartile range below the first quartile 
or more than times the interquartile range above the third quartile 
, where the interquartile range is defined as 
. How many outliers does this data set have?
Solution
Question 6
The circle having 
and 
as the endpoints of a diameter intersects the -axis at a second point. What is the -coordinate of this point?
Solution
Question 7
The functions 
and 
are periodic with least period 
. What is the least period of the function 
?
It's not periodic.
Solution
Question 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
Solution
Question 9
A circle has center 
and radius 
. Another circle has center 
and radius 
. The line passing through the two points of intersection of the two circles has equation 
. What is 
?
Solution
Question 10
At Typico High School, 
of the students like dancing, and the rest dislike it. Of those who like dancing, 
say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 
say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
Solution
Question 11
Call a positive integer 
if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 
, 
, and 
are monotonous, but 
, 
, and 
are not. How many monotonous positive integers are there?
Solution
Question 12
What is the sum of the roots of 
that have a positive real part?
Solution
Question 13
In the figure below, of the disks are to be painted blue, are to be painted red, and 
is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
Solution
Question 14
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
Solution
Question 15
Let 
be an equilateral triangle. Extend side 
beyond 
to a point 
so that 
. Similarly, extend side 
beyond 
to a point 
so that 
, and extend side 
beyond 
to a point 
so that 
. What is the ratio of the area of 
to the area of 
?
Solution
Question 16
The number 
has over 
positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Solution
Question 17
A coin is biased in such a way that on each toss the probability of heads is 
and the probability of tails is 
. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
The probability of winning Game A is 
less than the probability of winning Game B.
The probability of winning Game A is 
less than the probability of winning Game B.
The probabilities are the same.
The probability of winning Game A is greater than the probability of winning Game B.
The probability of winning Game A is greater than the probability of winning Game B.
Solution
Question 18
The diameter 
of a circle of radius is extended to a point 
outside the circle so that 
. Point 
is chosen so that 
and line 
is perpendicular to line 
. Segment 
intersects the circle at a point between and . What is the area of 
?
Solution
Question 19
Let 
be the 
-digit number that is formed by writing the integers from to in order, one after the other. What is the remainder when 
is divided by 
?
Solution
Question 20
Real numbers and are chosen independently and uniformly at random from the interval 
. What is the probability that 
, where 
denotes the greatest integer less than or equal to the real number 
?
Solution
Question 21
Last year Isabella took 
math tests and received different scores, each an integer between 
and 
, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 
. What was her score on the sixth test?
Solution
Question 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
Solution
Question 23
The graph of 
, where 
is a polynomial of degree , contains points 
, 
, and 
. Lines , 
, and 
intersect the graph again at points , , and 
, respectively, and the sum of the -coordinates of , , and is 
. What is 
? 
Solution
Question 24
Quadrilateral 
has right angles at and , 
, and 
. There is a point in the interior of such that 
and the area of 
is times the area of 
. What is 
?
Solution
Question 25
A set of 
people participate in an online video basketball tournament. Each person may be a member of any number of -player teams, but no teams may have exactly the same members. The site statistics show a curious fact: The average, over all subsets of size 
of the set of participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size 
of the set of participants, of the number of complete teams whose members are among those people. How many values , 
, can be the number of participants?
Solution
Answer Keys
Question 1: E
Question 2: E
Question 3: D
Question 4: C
Question 5: B
Question 6: D
Question 7: B
Question 8: C
Question 9: A
Question 10: D
Question 11: B
Question 12: D
Question 13: D
Question 14: E
Question 15: E
Question 16: B
Question 17: D
Question 18: D
Question 19: C
Question 20: D
Question 21: E
Question 22: B
Question 23: D
Question 24: D
Question 25: D