Sophomore Olympiad 2006
1. If n is a real number, then equals:
A) 1
B) 2
C)
D)
E) none of these
2. The odd number is:
A) 599
B) 1197
C) 1199
D) 1201
E) none of these
3. The number is:
A) even
B) odd
C) prime
D) multiple of 6
E) none of these
4. The domain of the function is
A)
B)
C)
D)
E) none of these
5. The range of the function is:
A)
B)
C)
D)
E) none of these
6. The radius of the circle that passes through the points (0,0), (1,0), and (0,1) is equal to:
A)
B)
C)
D) a circle that passes through the given points does not exist
E) none of these
7. The inverse,, of the function is:
A)
B)
C)
D)
E) none of these
8. If the graph of passes through the points (0,1) and (1,1), then equals:
A) 1
B) 2
C)
D)
E) none of these
9. The sum of the values of for which the two lines of equations and are perpendicular is:
A) -1
B) 0
C) 1
D) 2
E) none of these
10. If for all x, then equals:
A)
B)
C)
D) impossible to determine
E) none of these
11. Three coins are drawn at once from a jar containing two half dollars, four quarters, two dimes, and two nickels. The probability of drawing sixty cents is:
A)
B)
C)
D)
E) none of these
12. The sum of all solutions to the equation is:
A) -1
B) 1
C) 5
D) the equation has no solutions
E) none of these
13. The sum of the solutions of the equation is:
A)
B) 0
C) 2
D)
E) none of these
14. Let ABC be a right triangle with mA = . If we consider three squares, square
one has side length AB, square two has side length AC, and square three has side length BC, then:
A) area of square three – area of square two < area of square one
B) area of square three – area of square one > area of square two
C) area of square three – area of square two = area of square one
D) the relation between the areas depends on the lengths of the sides of the triangle
E) none of these
15. The solution set for is:
A)
B)
C)
D)
E) none of these
16. The product of the values of the parameter m for which the line is tangent to the circle of center (0,0) and radius 2 is:
A)
B)
C)
D)
E) none of these
17. If and , then is equal to:
A)
B)
C)
D)
E) none of these
18. The solution set for the inequality is:
A)
B)
C)
D)
E) none of these
19. Let m be an arbitrary real number. The solution set of the quadratic inequality is:
A)
B)
C) the empty set
D)
E) none of these
20. The solution set for the inequality is:
A) the empty set
B)
C)
D)
E) none of these
21. The sum of the solutions of the equation is:
A) -3
B) 0
C) 1
D) 3
E) none of these
22. The solution set for the inequality is:
A)
B)
C) the empty set
D)
E) none of these
23. Let ABC be a triangle with D a point on the side AB such that mABC = mACD. If AD = 2 cm and BD = 8 cm, then AC in cm is:
A)
B)
C)
D) 6
E) none of these
24. Given three arbitrary points that are not on the same line, then:
A) there is not necessarily a circle that passes through all three points
B) there is more than one circle that passes through all three points
C) there is only one circle that passes through all three points
D) there is only one circle that passes through any two points
E) none of these
25. If 1, -1 , and 1+ i are zeroes of a real polynomial P(x), then the most accurate conclusion about the degree of the polynomial is:
A) degree = 3
B) degree 3
C) degree = 4
D) degree 4
E) none of these
26.Wal-Mart is having a “25% off the price marked” on items that are already marked 50% off. The total discount on the original price of these items is:
A) 62.5%
B) 65.5%
C) 70%
D) 75%
E) none of these
27.If a polynomial changes its signs five times over, then the number of real zeros of is:
A)
B)
C)
D) not enough information to determine
E) none of these
28.Consider a semicircle with diameter and center . The line through perpendicular to meets the semicircle at , and is the midpoint of . The tangent at to the semicircle meets line at . If line crosses the semicircle at , then the area of the region is:
A)
B)
C)
D)
E) none of these
29. Let denote the gross monthly sales of a small company. If the manager receives a fixed salary of $4000 a month plus a commission rate of 5% on the monthly sales that exceed $50,000, then a function that represents the managers monthly salary is:
A)
B)
C)
D)
E) none of these
30.A real polynomial has exactly 4 real zeros and an absolute minimum. A possible equation of is:
A)
B)
C)
D)
E) none of these
31.The number of irreducible factors contained in the factorization over the real numbers of is:
A)
B)
C)
D)
E) none of these
32.Two circles of centers A and B and radii and , respectively, are externally tangent. Let be a line segment that is tangent to one circle at and the other at . The area of the region is:
A)
B)
C)
D)
E) none of these
33.If , then where is equal to:
A)
B)
C)
D)
E) none of these
34.The number of distinct 8-letter permutations of the letters in the word PARALLEL is:
A)
B)
C)
D)
E) none of these
35.The statement “If some Vulcans are Cardasians and some Cardasians are from the delta quadrant, then some Vulcans must be from the delta quadrant” is:
A) true
B) false
C) is such that the validity cannot be determined
D) neither true nor false
E) none of these
36.Let be a triangle with . If and are two points on such that , then the triangle must be:
A) isosceles but not equilateral
B) equilateral
C) triangle
D) triangle
E) none of these
37.Assume that your grade on algebra tests is inversely proportional to how many hours you sleep the night before the test. Assume also that your grade is when you sleep hours the night before the test. If you get hours of sleep on the night before the test, then your grade is:
A)
B)
C)
D)
E) none of these
38.If the area of the circular base of a right circular cone is quadrupled and the resulting cone is similar to the original cone, then the volume of the larger cone is:
A) times the volume of the original cone
B) times the volume of the original cone
C) 8 times the volume of the original cone
D) 16 times the volume of the original cone
E) none of these
39.If it takes ten workers to paint houses in days, then the time it takes five workers to paint houses is:
A) days
B) days
C) days
D) days
E) none of these
40. Let . If and , then is:
A)
B)
C)
D)
E) none of these
41.Missy is the 50th best and the 50th worst student in her class. The number of students in Missy’s class is:
A)
B)
C)
D)
E) none of these
42.Given that is a root for the polynomial, then the sum of all real roots of is:
A)
B)
C)
D)
E) none of these
43.If is a quadrilateral such that its diagonals are angle bisectors and perpendicular to each other, then the quadrilateral is:
A) a rhombus
B) a rectangle
C) a square
D) all of the above
E) none of these
44.The remainder of is:
A)
B)
C)
D)
E) none of these
45.Let be the center of two concentric circles with radii of 1 cm and 2 cm, and let the measure of angle be . The area of the region between the two circles in the interior of the angle is:
A)
B)
C)
D)
E) none of these
46.Let be the set . Thenequals:
A)
B)
C)
D)
E) none of these
47.The expression equals:
A)
B)
C)
D)
E) none of these
48.A breakfast cereal company produces a brand of cereal with a stated net weight of Only boxes with a net weight within of the stated amount are acceptable. Among the following, the box that is of acceptable weight is:
A) a box with a weight of
B) a box with a weight of
C) a box with a weight of
D) a box with a weight of
E) none of these
49.The graph of is moved to the right units and lifted up units. The equation of the new graph is:
A)
B)
C)
D)
E) none of these
50. The average on a math test was . Feeling bad, the teacher raised the grade of every student by points. The new average and standard deviation result in the following:
A) they are the same as the original ones
B) they are higher than the original ones
C) the average is the same, but the standard deviation is higher.
D) the average is higher, and the standard deviation remains the same
E) none of these