Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Junior Olympiad 1. If f x 2 x 3 Ax 2 Bx 5 , f 2 3 , and f 2 37 , then the value of A B is A) -14 B) -1 C) 1 2 D) 20 E) none of these 4 x if x 2 2. If f x , then the value of f 1 f 4 is x 2 if 2 x A) -6 B) 0 C) 6 D) 10 E) none of these 3. The coordinates of the point on the line 3x 2 y 0 that is equidistant from 0,0 and 2,3 is 3 A) ,1 2 3 B) 1, 2 3 C) ,1 2 3 D) 1, 2 E) none of these 4. If the point 2,5 is at a distance of 65 from the midpoint of the segment joining 4,2 and x,4 , then the value of x is A) 2 B) 2 C) 2 D) 2 E) none of these 5. Using interval notation, the solution set for the inequality A) 6,4 x 3 is x4 B) ,4 6, C) 6,4 D) ,4 6, E) none of these 6. Using interval notation, the solution set for 6 3x 7 10 is 11 A) 1, 3 B) , 17 C) , 3 11 D) ,1 , 3 E) none of these 7. Given the arithmetic sequence 5, 16, 27, 38, 49, …, the 138th number in the sequence is A) 1372 B) 1378 C) 1512 D) 1518 E) none of these 8. A student tosses a 6-sided die twice. The probability that the sum is odd and no more than 5 is 1 A) 36 B) 1 9 C) 1 6 D) 1 2 E) none of these 9. A student draws two cards without replacement from a regular fair deck. The probability that the second is black, given that the first is a spade is 1 A) 25 B) 25 52 C) 25 51 D) 1 2 E) none of these 10. If n A 54 , n A B 22 , n A B 85 , n A B C 4 , n A C 15 , nB C 16 , nC 44 , and nB' 63 , then nB C is A) 39 B) 81 C) 108 D) 118 E) none of these 11. If n is any real number, then 2 n 2 n is equal to A) 2 n 1 B) 4 n C) 2 2 n D) 4 2 n E) none of these 12. The number of times the digit nine occurs in the numbers from 1 to 100 is A) 10 B) 12 C) 19 D) 20 E) none of these 13. The equation of a line is given by x y 5 . The acute angle that the line makes with the x -axis is 2 A) 2 B) 4 C) 3 D) 2 E) none of these 14. 2 4 3 is equal to A) 8 6 B) 6 6 C) 5 6 D) 4 12 E) none of these 15. If n is any real number, then the number below that is not a perfect square is A) 2 2 n B) 9 n C) 5 6 n 2 D) 312 n E) none of these 16. The number of real solutions of x 3 3 x 1 is A) none B) one C) two D) three E) none of these 17. If h 0 and f x x 1 , then A) 2h h B) 2x h 2 h C) 1 D) 1 E) none of these f x h f x is equal to h x2 2 18. Where defined, the expression x 4 is equal to 2x 2 1 x x4 A) 4 B) C) D) x 2 x 4 x x 4x 2 2 x2 2 8 x 4x x4 E) none of these 19. The graph that represents a population of a city that increases at a decreasing rate is C) A) D) B) E) none of these 20. Let M be the midpoint of AB in a triangle ABC with mA 25 . If CM MB , then mB is equal to A) 45 B) 55 C) 65 D) 75 E) none of these 21. United States imports increased by 20% and exports decreased by 10% during a certain year. The ratio of imports to exports at the beginning of the year was x. The ratio of imports to exports at the end of the year is 11 x A) 8 B) 4 x 3 C) 3 x 2 D) 2x E) none of these 22. Let ABC be an isosceles triangle with base BC . Let BM and CN be the perpendiculars from B and C to AC and AB , respectively. If AB 5 cm and AM 4 cm , then NC is A) 3 cm B) 10 cm C) 4 cm D) 5 cm E) none of these 23. Let ABC be an isosceles triangle with base BC . If BH is the perpendicular from B to AC , then CBH is congruent to A) A B) A 2 C) A 3 D) not enough information to tell E) none of these 24. Let f be a function over the negative real numbers. If f then f x is 1 A) x 1 x 1 x for all x 0 , B) x C) x D) 1 x E) none of these 25. If you can move your finger up, down, left, or right only, then the number of 4digit telephone number extensions that start with 2 that you can dial is A) 1800 B) 2000 C) 2800 D) 10000 E) none of these 26. The solution set for the inequalities 4 x 2 9 is A) 2 x 3 B) 3 x 3 C) 3 x 2 2 x 3 D) 3 x 2 x 3 E) none of these 27. The number of different vertical arrangements of 8 flags (4 whites, 2 blues and 2 reds) is A) 16 B) 20 C) 70 D) 420 E) none of these 28. If x 1 and y 2 , then A) xy 2 B) x 1 y 2 C) y x 1 D) x y 1 E) none of these 29. If x 0 , then 413 log2 A) 4 x equals 3 x2 B) 12 x C) 6x 2 D) 4x 3 E) none of these 30. If k is a real number, then the set of values of k for which the graph of x 2 kx does not intersect the graph of y 1 is A) 2 k 2 B) k 2 k 2 C) k 1 D) the empty set E) none of these 31. Let C be a semi-circle centered at the origin O and diameter AB 4 cm. Let P be a point in the second quadrant on C . The arc AP , for which the area of OPB is 3 cm 2 , has length (in cm) A) 4 B) 3 C) 2 3 D) 5 6 E) none of these 32. The price P (in dollars) and the quantity x sold of a certain product obey the demand equation Px 5x 160 , 0 x 32 . The price that maximizes the company’s revenue is A) $16 B) $24 C) $19.2 D) $32 E) none of these 33. The faculty at a college consists of 80 full-time teachers and 52 part-time teachers. Of the 80 full-time teachers, 40 are female. Of the 52-part time teachers, 26 are female. The probability that a randomly selected teacher is male or works part time is 13 A) 66 B) 1 2 C) 23 33 D) 59 66 E) none of these 34. The expression A) B) C) 49 x 1 2 2 3x 49 x 49 x 2 2 2 1 2 equals 3x 2 49 x 3 2 2 49 2 x 2 49 x 3 2 2 49 x 2 49 x 3 2 2 3x 2 D) 49 x 2 E) none of these 35. If the probability of having a girl or a boy is the same, then the probability of a 4child family having 4 girls is 1 A) 32 B) 1 16 C) 1 8 D) 1 4 E) none of these 3 7 36. The expression 13 x 9 x 5 8 A) x 7 13 x 7 9 x 5 7 equals 3 8 8 B) x 7 13 x 7 9 x 9 8 C) x 13 x 7 9 5 7 8 D) x 13 9 x 9 1 3 E) none of these 37. Missy can wallpaper 2 rooms in a new house in 10 hours. Together with her trainee they can wallpaper the 2 rooms in 5 hours. By herself, the trainee can do the job in A) 10 hours B) 12 hours C) 22 hours D) 24 hours E) none of these 38. The domain of the real-valued function g x A) 49, B) 7,7 C) ,7 7, D) all real numbers except 7 and -7 E) none of these x x 2 49 is 39. The sum of the solution(s) to the equation log 4 x 4 log 4 x 2 2 is A) 6 B) 2 C) 4 D) 5 E) none of these 40. In a survey of 462 computer buyers, 256 put price as a main consideration, 211 put performance as a main consideration, and 56 listed both price and performance. The number of computer buyers who listed performance only as a main consideration is A) 51 B) 107 C) 155 D) 211 E) none of these 41. The owners of a candy store want to sell, for $6 per pound, a mixture of chocolate-covered raisins, which sells for $3 per pound, and chocolate-covered macadamia nuts, which sells for $8 per pound. They have a 40-pound barrel of raisins. The number of pounds of the nuts they should mix with the barrel of raisins so that they hit their target value of $6 per pound for the mixture is A) 52 B) 56 C) 60 D) 64 E) none of these 42. If f x f 1 x3 and x 2 , then g x is x x x g A) x 1 x3 B) x3 x 1 C) x 1 x3 D) x3 x 1 E) none of these 43. If f x 1 , then f f f x is x A) 1 x B) 1 x3 C) x D) 1 x2 E) none of these 44. If x100 x 1 is divided by x 1 , then the remainder is A) 1 B) 1 C) 2 D) 3 E) none of these 45. The value of m that makes the two lines of equations 3x y 2 and mx 2 y 1 perpendicular to each other is A) 2 3 B) 1 3 C) 1 3 D) 2 3 E) none of these 46. The vertex of the parabola that passes through the points 0,1, 1,0, 2,3 is 3 1 A) , 4 8 3 B) ,10 2 C) 1,1 1 D) 1, 2 E) none of these 47. A value of m that makes the graph of f x x 4 m 2 1 cross the x-axis is A) 4 B) 1 C) 0 D) does not exist E) none of these 48. A value of m that makes the graph of f x x 3 x m have no x-intercepts is A) 0 B) 1 C) 100 D) does not exist E) none of these 49. Among the following, the one that is a true identity is x4 x 2 A) B) ln x y ln x ln y C) e x y e x e y D) 1 1 1 x y x y E) none of these 50. If f 2x 1 x 4 for all x, then f x is A) 5 B) x 1 2 C) x2 2 D) x9 2 E) none of these