Question 1

What is the difference between the sum of the first even counting numbers and the sum of the first odd counting numbers?

Solution

Question 2

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?

Solution

Question 3

A solid box is cm by cm by cm. A new solid is formed by removing a cube cm on a side from each corner of this box. What percent of the original volume is removed?

Solution

Question 4

It takes Mary minutes to walk uphill km from her home to school, but it takes her only minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

Solution

Question 5

The sum of the two 5-digit numbers and is . What is ?

Solution

Question 6

Define to be for all real numbers and . Which of the following statements is not true?

for all and

for all and

for all

for all

if

Solution

Question 7

How many non-congruent triangles with perimeter have integer side lengths?

Solution

Question 8

What is the probability that a randomly drawn positive factor of is less than ?

Solution

Question 9

A set of points in the -plane is symmetric about the origin, both coordinate axes, and the line . If is in , what is the smallest number of points in ?

Solution

Question 10

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of , respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?

Solution

Question 11

A square and an equilateral triangle have the same perimeter. Let be the area of the circle circumscribed about the square and the area of the circle circumscribed around the triangle. Find .

Solution

Question 12

Sally has five red cards numbered through and four blue cards numbered through . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

Solution

Question 13

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

Solution

Question 14

Points and lie in the plane of the square such that , , , and are equilateral triangles. If has an area of 16, find the area of .

Solution

Question 15

A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

Solution

Question 16

A point P is chosen at random in the interior of equilateral triangle . What is the probability that has a greater area than each of and ?

Solution

Question 17

Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with radius and center at points and . What is the distance from to ?

Solution

Question 18

Let be a -digit number, and let and be the quotient and the remainder, respectively, when is divided by . For how many values of is divisible by ?

Solution

Question 19

A parabola with equation is reflected about the -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of and , respectively. Which of the following describes the graph of ?

Solution

Question 20

How many -letter arrangements of A's, B's, and C's have no A's in the first letters, no B's in the next letters, and no C's in the last letters?

Solution

Question 21

The graph of the polynomial

has five distinct -intercepts, one of which is at . Which of the following coefficients cannot be zero?

Solution

Question 22

Objects and move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object starts at and each of its steps is either right or up, both equally likely. Object starts at and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

Solution

Question 23

How many perfect squares are divisors of the product ?

Solution

Question 24

If what is the largest possible value of

Solution

Question 25

Let . For how many real values of is there at least one positive value of for which the domain of and the range of are the same set?

Solution

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