Question 1
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took hours and minutes, with , what is ?
Solution
Question 2
Which of the following is equal to ?
Solution
Question 3
What number is one third of the way from to ?
Solution
Question 4
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
Solution
Question 5
One dimension of a cube is increased by , another is decreased by , and the third is left unchanged. The volume of the new rectangular solid is less than that of the cube. What was the volume of the cube?
Solution
Question 6
Suppose that and . Which of the following is equal to for every pair of integers ?
Solution
Question 7
The first three terms of an arithmetic sequence are , , and respectively. The th term of the sequence is . What is ?
Solution
Question 8
Four congruent rectangles are placed as shown. The area of the outer square is times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
Solution
Question 9
Suppose that and . What is ?
Solution
Question 10
In quadrilateral , , , , , and is an integer. What is ?
Solution
Question 11
The figures , , , and shown are the first in a sequence of figures. For , is constructed from by surrounding it with a square and placing one more diamond on each side of the new square than had on each side of its outside square. For example, figure has diamonds. How many diamonds are there in figure ?
Solution
Question 12
How many positive integers less than are times the sum of their digits?
Solution
Question 13
A ship sails miles in a straight line from to , turns through an angle between and , and then sails another miles to . Let be measured in miles. Which of the following intervals contains ?
Solution
Question 14
A triangle has vertices , , and , and the line divides the triangle into two triangles of equal area. What is the sum of all possible values of ?
Solution
Question 15
For what value of is ?
Note: here .
Solution
Question 16
A circle with center is tangent to the positive and -axes and externally tangent to the circle centered at with radius . What is the sum of all possible radii of the circle with center ?
Solution
Question 17
Let and be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is , and the sum of the second series is . What is ?
Solution
Question 18
For , let , where there are zeros between the and the . Let be the number of factors of in the prime factorization of . What is the maximum value of ?
Solution
Question 19
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were and , respectively. Each polygon had a side length of . Which of the following is true?
Solution
Note that side is 2, and midpoint is 1. Therefore, square of R = square of r + 1.
Therefore, area of outer circle - area of inner circle = pie * square of R - pie * square of r = pie
This is true for any polygon. Therefore, A=B.
Question 20
Convex quadrilateral has and . Diagonals and intersect at , , and and have equal areas. What is ?
Solution
Question 21
Let , where , , and are complex numbers. Suppose that
What is the number of nonreal zeros of ?
Solution
Question 22
A regular octahedron has side length . A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. What is ?
Solution
Question 23
Functions and are quadratic, , and the graph of contains the vertex of the graph of . The four -intercepts on the two graphs have -coordinates , , , and , in increasing order, and . The value of is , where , , and are positive integers, and is not divisible by the square of any prime. What is ?
Solution
Question 24
The tower function of twos is defined recursively as follows: and for . Let and . What is the largest integer such that
is defined?
Solution
Question 25
The first two terms of a sequence are and . For ,
What is ?
Solution
Answer Keys
Question 1: A
Question 2: C
Question 3: B
Question 4: A
Question 5: D
Question 6: E
Question 7: B
Question 8: A
Question 9: D
Question 10: C
Question 11: E
Question 12: B
Question 13: D
Question 14: B
Question 15: D
Question 16: D
Question 17: C
Question 18: B
Question 19: C
Question 20: E
Question 21: C
Question 22: E
Question 23: D
Question 24: E
Question 25: A