Question 1

Kate bakes a -inch by -inch pan of cornbread. The cornbread is cut into pieces that measure inches by inches. How many pieces of cornbread does the pan contain?

Solution

---

Question 2

Sam drove miles in minutes. His average speed during the first minutes was mph (miles per hour), and his average speed during the second minutes was mph. What was his average speed, in mph, during the last minutes?

Solution

---

Question 3

In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?

Solution

---

Question 4

A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are and square units. What is ?

Solution

---

Question 5

How many subsets of contain at least one prime number?

Solution

---

Question 6

A box contains chips, numbered and . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds . What is the probability that draws are required?

Solution

---

Question 7

In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is . What is ?

Solution

---

Question 8

Sara makes a staircase out of toothpicks as shown:

This is a -step staircase and uses toothpicks. How many steps would be in a staircase that used toothpicks?

Solution

---

Question 9

The faces of each of standard dice are labeled with the integers from to . Let be the probability that when all dice are rolled, the sum of the numbers on the top faces is . What other sum occurs with the same probability ?

Solution

---

Question 10

In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?

Solution

---

Question 11

Which of the following expressions is never a prime number when is a prime number?

Solution

---

Question 12

Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

Solution

---

Question 13

How many of the first numbers in the sequence are divisible by ?

Solution

---

Question 14

A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?

Solution

---

Question 15

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?

Solution

---

Question 16

Let be a strictly increasing sequence of positive integers such that What is the remainder when is divided by ?

Solution

---

Question 17

In rectangle , and . Points and lie on , points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?

Solution

---

Question 18

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?

Solution

---

Question 19

Joey, Chloe and their daughter Zoe all have the same birthday. Joey is year older than Chloe, and Zoe is exactly year old today. Today is the first of the birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Solution

---

Question 20

A function is defined recursively by and for all integers . What is ?

Solution

---

Question 21

Mary chose an even -digit number . She wrote down all the divisors of in increasing order from left to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?

Solution

---

Question 22

Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?

Solution

---

Question 23

How many ordered pairs of positive integers satisfy the equation where denotes the greatest common divisor of and , and denotes their least common multiple?

Solution

---

Question 24

Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?

Solution

---

Question 25

Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?

Solution

Answer Keys

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