2. Use a calculator to approximate the following: What does this expression represent?
3. If r is a real number, either positive, negative, or zero, which of the following is always greater than r? Why?
4. Six students each try to guess the number of pennies in a jar. The six guesses are 52, 59, 62, 65, 49, and 42. One guess is 12 away and the other guesses are 1, 4, 6, 9, and 11 away. How many pennies are in the jar?
6. Each of the circles has a 10 inch radius. ABCD is a square inscribed in a semi-circle; MNPQ is a square inscribed in a circle. What is the ratio of the area of ABCD to MNPQ?
7. If a circle and a square have the same perimeter, which of the following statements are true?A. Their areas are equal.B. The area of the circle is greater than the area of the square.C. The area of the square is greater than the area of the circle.D. The area of the circle is π times the area of the square.
8. Draw an example of an octagon with:A. One line of symmetryB. Two lines of symmetryC. Four lines of symmetryD. Eight lines of symmetry
9. A vertical line divides the triangle with vertices (0,0), (1,1) and (9,1) into two regions of equal area. What is the equation of the vertical line?
10. The radius of a cylinder is 8 inches. The height of the cylinder is 3 inches. What number of inches can be added to either the radius or the height to give the same positive increase in volume?
11. A circle with area 9π is inscribed in an equilateral triangle. What is the area of the triangle?
12. Let ΔABC be a right triangle with AC as hypotenuse. Let CAB be a circle of diameter AB. Let CBC be a circle of diameter BC. Let CAC be a circle of diameter AC. Show that area CAB + area CBC = area CAC
13. What is the 500th fraction in the sequence?What is the nth fraction in the sequence?
14. The point (3, 5) is the top vertex of an isosceles triangle with base parallel to the x-axis. The triangle has an area of 12. Find the coordinates of the other two vertices.
15. If the circumference of a pipe is increased from 20 inches to 25 inches, what is the increase in the radius?
16. An 8 x 4 rectangle is centered at (-3, 7). What are the coordinates of the four corner points of the rectangle is the longer side is parallel to the y-axis?
17. In this figure, MNOP is a square and DAN is an equilateral triangle. If the area of MNOP is one square inch, what is the area of DAN in square inches?
18. Given a regular octagon with sides of length 8, find the distance between opposite sides.
19. Given f(x) = ax2 + bx + 5. Find a and b such that f(x + 1) – f(x) = 8x + 3
20. A wheel has an outside diameter of 6 feet. How many revolutions of the wheel are required to cause a point on the rim to go one mile?
21. Two cars start together and move in the same direction around a circular race track of length 999 meters. The cars meet for the first time 37 minutes after they start. Determine the speed of each car if the speed of the first car is 4 times the speed of the second.
22. A rectangular box measures 4 feet by 4 feet by 3 feet. Will a 6-foot pole fit inside?Justify your answer!
23. Solve the problems below. Each letter represents a unique digit. Letter values differ in the two problems.
24. The graph of a parabola that is symmetric about a vertical line contains the points (1, 4), (3, 18), and (0, 3). Find the equation of the parabola.
25. Describe and draw the solid that results from revolving a right triangle about one of its legs.
26. The odometer of the family car showed 15,951 miles. The driver noticed that this number was palindromic, that is, it read the same forward or backward. Two hours later, the odometer showed a new palindromic number. How fast was the car traveling in those two hours?
27. The function f(x) = x3 - 6x2 + 9x has a relative extremum at x = 3. Is it a maximum or a minimum?
28. ABCD and EFGH are squares, each with a side of length 12. E is the center of ABCD. If the distance BJ is 3, find the area of EJCK.
29. A fox notices a rabbit that is 30 meters away. The fox’s leap is 2 meters whereas the rabbit’s leap is 1 meter. If the fox can take 3 leaps in the time that the rabbit can take 2 leaps, what distance will the fox have to cover to catch the rabbit?
30. Find the points of intersection of the two graphs given by the equations 4x2 + y2 = 100 and 9x2 – y2 = 108
32. n and k are integers. What is the smallest value for n such that n3 – n is not divisible by k for all 1 ≤ k ≤ n?
33. Find the volume of a sphere circumscribed about a 4 cm by 4 cm by 4 cm cube.
34. Color the vertices of the given figure with as many colors as needed such that no two adjacent vertices are colored the same color. What is the minimum number of distinct colors needed?
35. If (2137)753 is multiplied out, the ones digit in the final product will be ________.
36. If the length of the diagonal of a square is y + w, what is the area of the square?
37. Three circles with centers (-2, 3), (0, 4), and (4, 6) are all tangent to each other at the point (2, 5). Find an equation for each circle.
39. The medians of a triangle meet at the centroid. Find the centroid of triangle ABC if A is (1, -2), B is (-2, 4), and C is (1, 6)
40. A tennis ball is served from a height of 7 feet to just clear a net 3 feet high. If it is served from a line 39 feet behind the net and travels in a straight path, how far from the net does the ball first hit the ground?
41. For what values of x will the graph of y = x2 – x – 2 be below the graph of y = x + 1?
42. The dot product of two vectors is defined as (a1, a2, a3 )(b1, b2, b3 ) = a1 b1 + a2 b2 + a3 b3. Find the dot product of (7, 6, 3) and (2, -1, 6).
43. Suppose a cable fits tightly around the equator of the earth. If an additional piece is to be spliced into the cable so that it can be raised three meters above the earth at all points, how much additional cable will be needed?
44. The altitudes of a triangle are concurrent at the orthocenter. Find the orthocenter of triangle ABC if A is (-2, 2), B is (4, 3), and C is (3, -2)
45. If f(x) is the inverse of g(x) under function composition, what is the relation of the graph of f(x) to the graph of g(x)?
46. If [a] denotes the greatest integer less than or equal to a, what is the value of [a] + [-a] when a is a real number?
48. When an ice cube melts until its sides are one third their original length, what fractional part of the ice cube has melted away?
49. Let A(-3, 1), B(2, 3), and C(-2, 5) be three points. If line CX is perpendicular to line AB at the point X, find the coordinates of X.
50. How far apart are the two circles x2 + y2 + 4x – 12y + 31 = 0 and x2 + y2 - 10x – 2y + 10 = 0?
51. What are the x- and y- intercepts of the hyperbola given by the equation
52. The current in an electrical circuit follows the equation l = 5(sin 8.72t) where t is the time in seconds and l is the current in amperes. What is the maximum value of the current in this circuit?
54. If a > 1 and 0 < x < y, which is larger, loga x or loga y?
55. Two high school classes took the same math test. One class of 20 students made an average grade of 80%. The other class had an average grade of 70%. If the combined average was 74%, how many students were in the second class?
56. If f(x) is any periodic function with period p, thenA. m · f(x) has period __________B. f(x + m) has period __________C. f(mx) has period ___________
57. A regular hexagon is inscribed in a circle of radius 10 inches. What is the area of the hexagon?
58. The location of a cannon is given coordinates (0, 100). The cannon fires a projectile which follows the path . for x ≥ 0. What are the coordinates of the point where the projectile hits the positive x-axis?
59. A parachutist will land somewhere in the square. What is the probability that she will land in the shaded area?
60. A box contains 5 red marbles and 3 green marbles. Three marbles are selected one after the other without replacing them as they are selected. What is the probability that the third marble selected is green?
61. A kite escaped the hands of a child. The little boy watched as the kite rose 200 m, traveled 1000 m due northwest, dropped 100 m, and continued to travel 500 m northeast. Then it turned and traveled 1000 m to the southeast and finally dropped 100 m. How far was the kite from its starting place?
64. A circular race track has a 50-meter radius for one running lane and a 52-meter radius for the second running lane. Two runners are beginning a race. How much of a headstart should the outside runner be given?
65. How many pairs (a, b) of non-zero real numbers satisfy the equation ?
67. On a square piece of paper, draw the largest possible circle and cut it out. Inside the circle, draw the largest possible square and cut it out. The final square is what fraction of the original piece of paper?
68. Four girls bought a bike for $60. The first girl paid one half of the sum of the amounts paid by the other girls; the second girl paid one third of the sum of the amounts paid by the other girls; the third girl paid one fourth of the sum of the amounts paid by the other girls. How much money did the fourth girl pay?
69. What is the area of the triangle lying between the coordinate axes and the line 3x + 8y – 24 = 0?
70. A ladder is 16 feet long. The top of the ladder rests against a window of a two-story house that is built on level ground. The ladder makes an angle between 700 and 850 with the ground. What are the minimum and maximum heights from the ground to the window?
Editor's Notes
Rational; pi
Number theory; A
Guessing probability; 53
Graphing absolute value
Area inscribed square circle; 2/5
Square circle perimeter area; B
Octagon symmetry
Area; x = 3
Cylinder volume; 16/3
Area circle inscribed; 27sqrrt 3
Pythagorean theorem
Sequences; 1999/2001; (4n-1)/(4n+1)
Isosceles triangle area; many possible correct ex. (1, -1) (5, -1)
