Question 1
The median of the list is . What is the mean?
Solution
Therefore n=4. Then use n to replace the numbers and calculate the average, or mean, which is E.
Question 2
A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
Solution
Question 3
The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
Solution
Question 4
What is the maximum number for the possible points of intersection of a circle and a triangle?
Solution
Question 5
How many of the twelve pentominoes pictured below have at least one line of symmetry?
Solution
Question 6
Let and denote the product and the sum, respectively, of the digits of the integer . For example, and . Suppose is a two-digit number such that . What is the units digit of ?
Solution
Question 7
When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?
Solution
Question 8
Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?
Solution
Question 9
The state income tax where Kristin lives is levied at the rate of of the first of annual income plus of any amount above . Kristin noticed that the state income tax she paid amounted to of her annual income. What was her annual income?
Solution
Question 10
If , , and are positive with , , and , then is
Solution
Question 11
Consider the dark square in an array of unit squares, part of which is shown. The ???rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring is
Solution
Question 12
Suppose that is the product of three consecutive integers and that is divisible by . Which of the following is not necessarily a divisor of ?
Solution
Also, whenever n is the product of three consecutive integers, n is divisible by 6.
It also mentions that it is divisible by 7, so the number is definitely divisible by all the factors of 42.
In our answer choices, the one that is not a factor of 42 is 28.
Question 13
A telephone number has the form , where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find .
Solution
The last four digits GHIJ are either 9753 or 7531, and the other odd digit (1 or 9) must be A, B, or C. Since A + B + C = 9, that digit must be 1. Thus the sum of the two even digits in ABC is 8.
DEF must be 864, 642, or 420, which respectively leave the pairs 2 and 0, 8 and 0, or 8 and 6, as the two even digits in ABC. Only 8 and 0 has sum 8, so ABC is 810, and the required first digit is 8.
Question 14
A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
Solution
Question 15
A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is feet and each stripe is feet long. Find the distance, in feet, between the stripes?
Solution
Question 16
The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?
Solution
1/3 * [(m-10) + 5 + (m + 15)] = m, or m=10.
So the sum of the three numbers is 30.
Question 17
Which of the cones listed below can be formed from a sector of a circle of radius by aligning the two straight sides?
Solution
The curve line will form the circumference of the base. We can compute its length and use it to determine the radius.
The length of the curve line is 252/360 * 2 * pi *10 = 14 * pi. This is the circumference of a circle with radius 7.
Question 18
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
Solution
Question 19
Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?
Solution
Question 20
A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?
Solution
2000-2x = x 2^2, where x is the side of the isosceles triangle.
Question 21
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and altitude , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
Solution
Question 22
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .
Solution
Question 23
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
Solution
Question 24
In trapezoid , and are perpendicular to , with , , and . What is ?
Solution
Question 25
How many positive integers not exceeding are multiples of or but not ?
Solution
Answer Keys
Question 1: E
Question 2: C
Question 3: E
Question 4: E
Question 5: D
Question 6: E
Question 7: C
Question 8: B
Question 9: B
Question 10: D
Question 11: C
Question 12: D
Question 13: E
Question 14: A
Question 15: C
Question 16: D
Question 17: C
Question 18: D
Question 19: D
Question 20: B
Question 21: B
Question 22: D
Question 23: D
Question 24: B
Question 25: B