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Senior Olympiad 2007 x 2 1 for x 4 1. If f ( x) , then f f 2 is: 6 x 7 for x 4 A) B) C) D) E) –19 5 23 362 none of these 2. If f ( x) 2 x 1 1 for all real numbers x, then the range of f is: A) B) C) D) E) , 1 , 1 1, , none of these 3. The vertex of the parabola defined by y 2 x 2 x 1 is: 9 1 A) , 8 4 9 1 B) , 8 4 15 1 C) , 4 16 17 1 D) , 4 16 E) none of these 4. If x 4 x 3 x 2 1 is divided by x 2 1 , then the remainder is: A) B) C) D) E) 1 x 1 x 1 x2 x 1 none of these 5. The unit’s digit of 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! is: 1 A) B) C) D) E) 2 3 4 5 6 7 8 9 10 0 1 5 7 none of these 6. The area of the triangle bounded by the graphs of 4 x 3 y 0, 3x 4 y 25 and y 0 is: A) B) C) D) E) 25 3 75 8 25 2 50 3 none of these 7. If A 2007 2007 2007 2007 and B 2007 2007 2007 2007 , then A 2 B 2 is: A) B) C) D) E) 1 0 1 2 none of these 8. The coordinates of the vertices of a rectangle are (1, 1), (5, 1), (1, 7), and (5, 7). If a point P is randomly selected within the interior of the rectangle, then the probability that the x-coordinate of P is greater than the y-coordinate of P is: A) B) C) D) E) 1 4 1 3 1 2 1 none of these 9. If the sum of two numbers is 50 and the product of the same two numbers is 25, then the sum of the reciprocals of these two numbers is: A) B) C) D) E) 3 50 2 75 1250 none of these 10. The area of the polygon bounded by the graphs of 3x 4 y 6, 4 x 3 y 18, x = 0, and y = 0 is: A) B) C) D) E) 3 4 5 6 none of these 11. If A B C , C D E , E A F , B D F 100 , and A 8 , then F is: A) B) C) D) E) 46 48 56 58 none of these 2 1 12. The sum of the solutions of x 3 6 x 3 7 is: A) B) C) D) E) 6 6 342 343 none of these 13. The negation of “If x 0 , then x 2 0 ” is: A) B) C) D) E) If x 0 , then x 2 0 . If x 2 0 , then x 0 . x 0 or x 2 0 . x 0 and x 2 0 . none of these 14. The set of all (x, y) such that 2 x 2 y 2 4 x 4 y 1 is: A) B) C) D) E) the interior of a circle the exterior of a circle the interior of an ellipse the exterior of an ellipse none of these 15. The conic section defined by x 2 2 x 4 y 2 1 is: A) B) C) D) E) a circle a parabola a hyperbola an ellipse none of these 16. Let i 1 . If 1 3i is a zero of f ( x) x 4 3x 3 6 x 2 2 x 60 , then the sum of all the real zeros of f (x) is: A) B) C) D) E) 60 3 1 3 none of these 17. The number of solutions of ln x 2 ln 2x 3 2 ln x is: A) B) C) D) E) 0 1 2 3 none of these 18. If there are 100, 1000, and 10,000 elements in sets A, B, and C, respectively; two elements common to each pair of sets; and one element common to all three sets, then the number of elements in A B C is: A) B) C) D) E) 1 11,093 11,107 11,109 none of these 19. If 400 books are placed onto thirteen shelves, then at least one shelf must have at least: A) B) C) D) E) 31 books 32 books 33 books 34 books none of these 20. The number of subsets of {a, b, c, d, e} that do not contain the letter a is: A) B) C) D) E) 15 16 31 32 none of these 21. A fair coin is flipped eight times where each flip comes up either ‘heads’ or ‘tails’. The number of possible outcomes that contain exactly three heads is: A) B) C) D) E) 5 8 56 336 none of these 22. The number of permutations of the letters A, B, C, D, E, F, and G that contain the strings ABC and DE is: A) B) C) D) E) 24 35 840 5040 none of these 23. Where defined, the expression cot 2 sin 2 csc 2 equals: A) B) C) D) E) sin 2 sin 2 cos 2 cos 2 none of these 24. The sum, in degrees, of the solutions of 2 sin 2 3 cos in the interval 0 360 is: A) B) C) D) E) 60 120 240 360 none of these 25. The area of a parallelogram with an interior angle of measure 150 and two sides of length 8 and 26 is: A) B) C) D) E) 52 104 104 3 208 none of these 26. If the points 3, 22 , 1, 6 , and 0, 1 lie on the graph of y ax 2 bx c , then a b c equals: A) B) C) D) E) 2 6 27 29 none of these 27. The sum of the solutions of x 1 5x 1 6 0 equals: 4 A) B) C) D) E) 7 5 0 2 2 3 2 none of these 2 x y 1 28. If k is a constant such that has no solution (x, y), then k is: kx 2 y 2 A) B) C) D) E) 1 0 1 2 none of these 29. The domain of definition of the real-valued function defined by f ( x) A) B) C) D) E) , 4 4, , 4 4, 4, 4 4, 4, 4 x 2 16 is: x4 none of these 16 cm arc on a circle of radius 16 cm, 3 then the smaller angle, in degrees, between the tangent lines is: 30. If tangent lines are drawn from each end of a A) B) C) D) E) 30 60 90 120 none of these 31. The sum of all solutions of sin x sin x 1 in the interval 0 2 is: 3 3 A) B) C) D) E) 4 3 2 7 6 none of these 32. The sum of the solutions of sin 3 cos 1 in the interval 0 2 is: A) B) C) D) E) 6 2 5 3 7 3 none of these 33. If i 1 , then 1 i is: 12 A) B) C) D) E) 64 2 1 64i none of these 2 2 i sin 34. If i 1 , then the sum of the square roots of 4 cos is: 3 3 A) 2 2 3i B) C) D) E) 2 2 3i 0 2 none of these 35. The number of nonempty bit strings of length five or less is: A) B) C) D) E) 32 62 64 128 none of these 36. The number of functions from {1, 2, 3, 4} into {a, b, c, d} that do not map 1 onto a is: A) B) C) D) E) 18 256 456 512 none of these 37. The number of functions from {a, b, c, d} into {e, f, g, h} that are not one-to-one is: A) B) C) D) E) 24 64 232 256 none of these x2 y2 1 4 9 38. The number of solutions of is: 2 2 x 4 y 1 4 16 A) B) C) D) E) 0 1 2 4 none of these 39. The sum of the x and y - coordinates of the point on the graph of x 12 y 22 1 that is closest to the graph of x 7 2 y 62 4 is: A) B) C) D) E) 2 5 6 51 5 10 none of these 9 40. The number of real numbers x for which x i 1 is: i 1 A) B) C) D) E) 0 1 2 3 none of these 19 41. The value of 2 i is: i 1 A) B) C) D) E) 2 20 1 1 2 18 1 2 19 1 2 20 none of these 42. If f (3x 1) 4 x 2 for all x, then f (x) is: A) x 3 4x 3 B) 3 C) 7 x 3 4x 3 D) 10 E) none of these 43. The sum of all the coefficients in the expansion of x 2 y 100 A) B) C) D) E) – 2000 – 200 200 2000 none of these is: x 1 for x 0 44. If f ( x) , then f f (x) is: 2 x 1 for x 0 x 2 for x 1 A) 2 x 3 for 1 x 0 4 x 3 for x 0 x 2 2 x 1 for x 0 B) 2 4 x 4 x 1 for x 0 x 2 for x 0 C) 4 x 3 for x 0 2 x 1 for x 0 D) 4 x 2 for 0 x 1 4 x 1 for x 1 E) none of these 45. If tan x 3 and sin x 0 , then cos x equals: 4 4 5 3 5 3 5 4 5 none of these A) B) C) D) E) 46. If f ( x) x 2 and h 0 , then A) B) C) D) E) h 2a h 2 f 2a h none of these f ( a h) f ( a ) is equal to: h 2 47. If the x-intercepts of the graph of y x (a b) x ab are 5, 0 and 2, 0 , then a b is: A) B) C) D) E) 7 3 3 7 none of these 48. The expression A) B) C) D) E) 123456790 123456789 2 123456789 123456791 equals: 0 1 123456789 987654321 none of these 49. If x and y are positive integers such that y is the hundred’s digit of the six-digit integer 2 492y04 and 3x 690 492 y 04 , then x + y is: A) B) C) D) E) 6 12 14 15 none of these 50. A simple polygonal region is constructed by sequentially connecting the points 0, 2, 4, 3 , 6, 3 , 1, 1 , 2, 2, 4, 3 , and 0, 2 . The area of the region is: A) 24 67 B) 2 C) 48 D) 67 E) none of these