Question 1

Makarla attended two meetings during her -hour work day. The first meeting took minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?

Solution

Question 2

A big is formed as shown. What is its area?

Solution

Question 3

A ticket to a school play cost dollars, where is a whole number. A group of 9^th graders buys tickets costing a total of $, and a group of 10^th graders buys tickets costing a total of $. How many values for are possible?

Solution

Question 4

A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

Solution

Question 5

Lucky Larry's teacher asked him to substitute numbers for , , , , and in the expression and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for , , , and were , , , and , respectively. What number did Larry substitute for ?

Solution

Question 6

At the beginning of the school year, of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and answered "No." At the end of the school year, answered "Yes" and answered "No." Altogether, of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of ?

Solution

Question 7

Shelby drives her scooter at a speed of miles per hour if it is not raining, and miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of miles in minutes. How many minutes did she drive in the rain?

Solution

Question 8

Every high school in the city of Euclid sent a team of students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed ^th and ^th, respectively. How many schools are in the city?

Solution

Question 9

Let be the smallest positive integer such that is divisible by , is a perfect cube, and is a perfect square. What is the number of digits of ?

Solution

Question 10

The average of the numbers and is . What is ?

Solution

Question 11

A palindrome between and is chosen at random. What is the probability that it is divisible by ?

Solution

Question 12

For what value of does

Solution

Question 13

In , and . What is ?

Solution

Question 14

Let , , , , and be positive integers with and let be the largest of the sums , , and . What is the smallest possible value of ?

Solution

Question 15

For how many ordered triples of nonnegative integers less than are there exactly two distinct elements in the set , where ?

Solution

Question 16

Positive integers , , and are randomly and independently selected with replacement from the set . What is the probability that is divisible by ?

Solution

Question 17

The entries in a array include all the digits from through , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

Solution

Question 18

A frog makes jumps, each exactly meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than meter from its starting position?

Solution

Question 19

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than points. What was the total number of points scored by the two teams in the first half?

Solution

Question 20

A geometric sequence has , , and for some real number . For what value of does ?

Solution

Question 21

Let , and let be a polynomial with integer coefficients such that

, and
.

What is the smallest possible value of ?

Solution

Question 22

Let be a cyclic quadrilateral. The side lengths of are distinct integers less than such that . What is the largest possible value of ?

Solution

Question 23

Monic quadratic polynomials and have the property that has zeros at and , and has zeros at and . What is the sum of the minimum values of and ?

Solution

Question 24

The set of real numbers for which

is the union of intervals of the form . What is the sum of the lengths of these intervals?

Solution

Question 25

For every integer , let be the largest power of the largest prime that divides . For example . What is the largest integer such that divides

?

Solution

Answer Keys