2 5

ALABAMA
STATEWIDE MATHEMATICS CONTEST

DIVISION I COMPREHENSIVE EXAM

1. Suppose a and b are integers. Find a + b if (A) 4 (B) 3

11 6 2 = a + b 2. (C) 2 (D) 1 (E) 0

2. Which one of the following is not a factor of x6 1 ? (A) x 1 (B) x2 1 (C) x2 + x + 1 (D) x4 + x2 + 1 (E) All are factors

3. How many positive integers less than 70 are relatively prime to 70 ? (A) 22 (B) 24 (C) 26 (D) 28 (E) 30

4. What is the remainder if 22005 is divided by 13 ? (A) 0 (B) 1 (C) 2 (D) 4 (E) 7

5. How many consecutive zeros does (A) 8 6. If x + (A) (B) 10

100! have at the end ? 219 510 (C) 12 (D) 14 (E) 16

1 1 = 3 and x > 1, what is the value of x3 3 ? x x 8 5 (B) 5 2 (C) 8 5

(D) 5 2

(E) None of these

7. If 23a = 5, 23b = 10, what is the value of 2 ba ? (A) 5 (B) 10 (C) 15 (D) 20 (E) 23

8. Let f (x) = log (x + x2 + 1). Which one of the following statements are true ? I. The domain of f (x) is all real numbers. II. The graph of f (x) contains the origin (0, 0). III. f (x) = f (x). (A) I (B) II (C) I and II (D) II and III (E) I, II and III

1 1 and q = , what is p2 + pq + q 2 ? 9. If p = 14 13 14 + 13 (A) 49 (B) 52 (C) 55 (D) 58 (E) 61

10. What is the sum of the coecients of all the terms of (1 + 2x + 3x2 4x3 )10 when it is expanded ? (A) 4 (B) 16 (C) 256 (D) 1024 (E) 2048

t+1 11. If f ( t 3) = , what is f (2) ? t1 (A) 3 (B) 4 3 (C) 5 4 (D) 6 5 (E) 3

12. Find sum of the minimum distance and the maximum distance from the point (4, 3) to the circle x2 + y 2 + 4x 10y 7 = 0. (A) 10 (B) 15 (C) 20 (D) 25 (E) 30

13. If both of the functions f (x) = sin x and g(x) = cos x are dened on [0, ], what is 2 1 1 1 1 f f +g ? 3 3 (A) 1 (B) 1 3 1+i 1i (C)
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1 9

(D)

1 2

(E) None of these

14. If i =

1, what is the value of (B) 1

? (C) 1 (D) i (E) i

(A) 0

15. If the equality x3 = A(x 1)(x 2)(x 3) + B(x 1)(x 2) + C(x 1) + D is an identity, what is A+B+C +D ? (A) 13 (B) 15 (C) 17 (D) 19 (E) 21

16. If the height of a cylinder is increased by 10%, and the radius of the cylinder is decreased by 10%, what will happen to the volume of the cylinder ? (A) Same as the volume of the original cylinder (C) Decreases by 8.9% (D) Increases by 10.9% (B) Increases by 8.9% (E) Decreases by 10.9%

17. If f (x) = (A) 100

x2

x2 , what is 51 f (50) f (49) f (3) f (2) ? 1 (B) 120 (C) 140 x= (D) 160 (E) 180

18. Let x denote the following innite 1 1 1 y = 2 + 2 + 2 + in terms of x . 1 2 3 (A) 2x (B) 4 x 3

series:

1 1 1 + 2 + 2 + Express the series 2 1 3 5

(C)

5 x 3

(D)

6 x 5

(E)

7 x 5

19. Find B + 2C if the quadratic function f (x) = Ax2 + Bx + C has the minimum value 11 when x = 1 . (A) 22 (B) 20 (C) 18 (D) 16 (E) 14

20. Let O(0, 0), A(4, 13) and B(8, 9) be three points on the xy-plane. If the line y = kx bisects the area of the triangle OAB, what is k ? (A) 3 2 (B) 5 3 (C) 9 5 (D) 11 6 (E) None of these

10

21. Find the remainder when the sum

n2 + 3n + 1 n! is divided by 10 . (C) 5 (D) 7 (E) 9

(A) 1

(B) 3

22. Suppose that a function f (x) dened on natural numbers satises following conditions: (i) f (1) = 1 (ii) f (x + y) = f (x) + f (y) + xy for all x, y What is the value of f (1) + f (2) + f (3) + f (4) ? (A) 10 23. If sin16 a = (A) 4 (B) 15 (C) 20 (D) 25 (E) 30

1 1 2 4 1 , what is + + + ? 5 cos2 a 1 + sin2 a 1 + sin4 a 1 + sin8 a (B) 6 (C) 8 (D) 10 (E) None of these

24. There are seven points plotted on a semicircle as in the gure. How many triangles can be made by joining the points in the semicircle ? (A) 30 (B) 31 (C) 32 (D) 33 (E) None of these

25. How many solutions are there for the equation cos2 (x) sin2 (2x) = 0 on [0, 2] ? (A) 6 (B) 4 (C) 2 (D) 1 (E) None of these

26. Which ones of the following statements are true ? (ii) log2 (n + 2) > log3 (n + 2) . (i) log2 (n + 3) > log2 (n + 2) . (A) (i) (B) (ii) (C) (i) and (ii)

(iii) log2 (n + 2) > log3 (n + 3) . (E) (i),(ii) and (iii)

(D) (i) and (iii)

27. Imagine that you have two thumbtacks placed at two points, A and B. If the ends of a xed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse. What is the best way to maximize the area surrounded by the ellipse with a xed length of string? I Take the two points A and B apart so that they have the maximum distance. II Make two points A and B coincide. III Place A and B vertically. IV Place A and B horizontally. V The area is always same regardless of the location of A and B. (A) I (B) II (C) III (D) IV (E) V

28. Find the sum of the following series: 1 + (A) 7 3 (B) 5 2

1 1 1 1 1 1 1 + + + + + + + 2 3 4 9 8 27 16 (C) 7 4 (D) 5 3 (E) None of these

29. Suppose that 10 straight lines are drawn on a piece of paper so that every pair of lines intersects, but no three lines intersect at a common point. Into how many regions do the 10 straight lines divide the plane ? (A) 50 (B) 51 (C) 53 (D) 55 (E) 56
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30. Let and be the solutions to the equation 2x2 x 2 = 0. Find the value of 2 + 2 + () (A) 0 (B) 1 (C) 2 (D) 1

(E) 2

31. Michelle decides to start saving money. She plans to save 1 cent the rst day of January, 4 cents the second day, 10 cents the third day, 19 cents the fourth day, 31 cents the fth day, 46 cents the sixth day, and so on for the month. How much will she save in January? There are 31 days in January. (A) $13.96 (B) $14.71 (C) $135.16 (D) $149.11 (E) None of these

32. Nicole is making a bracelet putting beads on a string in order. If she has 10 dierent beads, how many dierent bracelets are possible ? (A) 9! 2 (B) 10! 2 (C) 9! (D) 10! (E) 11!

33. Among the following shapes of equal perimeter, which one has the largest area ? (A) circle (B) equilateral (C) square (D) ellipse (E) regular pentagon

34. How many three digit numbers are same when the rst and the last digit are interchanged ? (A) 1000 (B) 729 (C) 100 (D) 90 (E) 81

35. Eric wants to construct a conical paper cup by gluing together two sides AB AC in a semi circle and with radius, AB = AC = 2 3. What is the volume of the cup ? (A) (B) 2 3 (C) 4 3 (D) 3 (E) 9 36. Let x be 1 + 2+ 1+ 2+ 1+ (A) 1 (B) 5 1 1 1 1 1 2 + (C) 2

B=C 2 3 B A C A

. Evaluate (2x 1)2 .

(D) 7 2, f2 = log3 3 3, f3 = log4

(E)

3 4 4,

37. Dene the sequence of numbers as follows: f1 = log2 f4 = log5 5 5 5 5, . Find the n-th term fn . (A) 1 (B) logn+1 n+2 (C) 2n 1 2n

(D)

1 logn (n + 2)

(E) 2n

38. Let X = {1, 2, 3, 4}. How many functions f : X X are there satisfying (f f )(x) = x for all x in X ? (A) 10 (B) 11 (C) 12 (D) 13 (E) None of these

39. Find the area of the region consisting of all points (x, y) so that x2 + y 2 1 |x| + |y|. (A) (B) 1 (C) 2 (D) 3 (E) + 1

40. The following gure is obtained joining squares. Find the ratio of the side of the square A to the one of the square B. (A) 4:3 (B) 8:5 (C) 15:12 (D) 16:11 (E) 17:3

41. Holly wants to make a plan for January next year. She has a calendar for this year, but doesnt have one for next year. Which month of this year shows the same date and day as January of next year ? (A) January (B) March (C) May (D) August (E) October

42. Brian paid $50 for 12 of honeydews, coconut and watermelons. The unit price for honeydew, coconut and watermelon is $2, $5 and $9, respectively. Suppose that Brian bought at least one fruit for each. How many coconuts did Brian buy ? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

43. Suppose that two lines, y = ax and y = bx are symmetric with respect to the line y = x. If the angle between two lines is 20 , what is the value of 3ab ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

44. When the polynomial f (x) is divided by x + 3, the quotient is x2 1 and the remainder is 2. Find the remainder when this polynomial f (x) is divided by x 2 . (A) 1 (B) 2 (C) 15 (D) 17 (E) 20

45. Let a and b be the coecients of x3 in (1 + x + 2x2 + 3x3 )3 and (1 + x + 2x2 + 3x3 + 4x4 )3 , respectively. Find the value of a b . (A) 0 (B) 1 (C) 1 (D) 33 43 (E) 33 34

46. The line (k + 1)2 x + ky 2k 2 2 = 0 passes through a point regardless of the value k. Which of the following is the line with slope 2 passing through the point ? (A) y = 2x 8 (B) y = 2x 5 (C) y = 2x 4 (D) y = 2x + 5
5 x2

(E) y = 2x + 8

47. What percentage of the interval [10, 10] is inequality x + 2 > (A) 70% (B) 60% (C) 50%

satised ? (D) 40% (E) 30%

48. What is the radius of a circle which circumscribes the triangle with vertices (2, 0), (0, 4), and (4, 6) ? 5 (B) 10 (C) 1 (D) 2 5 (E) 40 (A) 49. Solve 3x + 9x = y for x. (A) x = 3 (B) x = ln y 2 ln 3 (C) x = ln y ln 3 ln 9 (D) x = ln y + 3 ln 9 (E) None of these

50. John tries to prove his conjecture that

f (a) is integer for two distinct integers, a and b for the given ab polynomial f (x) which satises the following two properties, (I) and (II). (I) Every coecient in f (x) is an integer . (II) f (a)f (b) = (a b)2 . It is well known that (III) For given integers m and n, if the solutions of a quadratic equation x2 + mx + n = 0 are rational numbers, then they are integers. Find out a mistake from his proof if there is any. Proof: (a) For a natural number n, f (a) f (b) is divisible by a b since an bn is divisible by a b by (I). f (a) f (b) f (a) f (b) is integer. The quadratic equation which has two solutions and is (b) Thus ab ab ab f (a) f (b) x2 x + 1 = 0 by (II). ab f (a) is rational by (I), and (c) Since ab f (a) f (b) f (a) (d) is integer, is integer by (III). ab ab (A) (a) (B) (b) (C) (c) (D) (d) (E) None