Counting and Probability
The Odle Middle School math team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
12
How many square yards of carpet are required to cover a rectangular floor that is 12 feet long and 9 feet wide? (There are 3 feet in a yard.)
12
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Aaron
Which of the following integers can't be written as the sum of four consecutive odd integers?
16, 40, 72, 100, 200
100
How many ways are there to choose positive integers a, b, c such that a < b < c and abc = 100?
4
How many integers between 1000 and 9999 have 4 distinct digits?
4536
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note that the diagram is not drawn to scale. What is the length of X, in centimeters?
5
Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?
000101
Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. What integer between 5 and 9 inclusive cannot be the sum of the two numbers?
6
Suppose a * b means 3a - b. What is the value of x if 2 * (5 * x) = 1?
10
How many different paths in the arrangement of letters and numbers below spell AMC8? Beginning at the A in the middle, a path may move to an orthogonally adjacent letter (i.e. left, right, up, down, not diagonal).
24
Rectangle ABCD is inscribed in a semicircle with diameter FE. Let DA = 16, and FD = AE = 9. What is the area of ABCD?
240
Let the letters B, F, G, L, U, Y represent distinct integers from 0 to 9 inclusive. Suppose FLYFLY is the greatest number satisfying the equation 8 * FLYFLY = BUGBUG
What is the value of FLY + BUG?
1107
How many digits are in the base-ten representation of 8^5 * 5^10 * 15^5?
18
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?
93
Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?
25/63
In the figure shown, US and UT are line segments of length 2, and <TUS = 60°. Arcs TR and SR are each one-sixth of a circle with radius 2. What is the area of the region SUTR?
4sqrt(3) - 4pi/3
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
146
What is the greatest integer less than or equal to (3^100 + 2^100)/(3^96 + 2^96)
80
A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and X. The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all possible values of X?
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that is tossed into bins numbered with positive integers so that for each ball, the probability that is tossed into bin k is 2^(-k) for k = 1,2,3,..... What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
1/3
The figure shown is octagon ABCDEFGH, consisting of rectangles and right triangles. When cut out and folded along these dotted lines, the polygon forms a triangular prism. Suppose AH = EF = 8, and GH = 14. What is the volume of the prism?
192
Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
103
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
729
The real numbers c, b, a form an arithmetic sequence with a >= b >= c >= 0. The quadratic ax^2 + bx + c has exactly one root. What is this root?
-2 + sqrt(3)