Question 1

What is the value of

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Question 2

Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?

Liliane has more soda than Alice.

Liliane has more soda than Alice.

Liliane has more soda than Alice.

Liliane has more soda than Alice.

Liliane has more soda than Alice.

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Question 3

A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

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Question 4

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

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Question 5

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?

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Question 6

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?

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Question 7

For how many (not necessarily positive) integer values of is the value of an integer?

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Question 8

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?

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Question 9

All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?

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Question 10

Suppose that real number satisfies What is the value of ?

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Question 11

When fair standard -sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as where is a positive integer. What is ?

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Question 12

How many ordered pairs of real numbers satisfy the following system of equations?

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Question 13

A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?

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Question 14

What is the greatest integer less than or equal to

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Question 15

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?

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Question 16

Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?

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Question 17

Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element in

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Question 18

How many nonnegative integers can be written in the form where for ?

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Question 19

A number is randomly selected from the set , and a number is randomly selected from . What is the probability that has a units digit of ?

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Question 20

A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?

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Question 21

Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?

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Question 22

Let and be positive integers such that , , , and . Which of the following must be a divisor of ?

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Question 23

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?

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Question 24

Triangle with and has area . Let be the midpoint of , and let be the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?

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Question 25

For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?

Solution

Answer Keys

Question 1: B
Question 2: A
Question 3: E
Question 4: E
Question 5: D
Question 6: B
Question 7: E
Question 8: C
Question 9: E
Question 10: A
Question 11: E
Question 12: C
Question 13: D
Question 14: A
Question 15: D
Question 16: D
Question 17: C
Question 18: D
Question 19: E
Question 20: B
Question 21: E
Question 22: D
Question 23: D
Question 24: D
Question 25: D