Download 2006 Exam

Junior Olympiad
1. The slope of any line parallel to the graph of 3 x  5 y  8 is:
A)  3
3
B) 
5
5
C)
3
D) 3
E) none of these
2. If x  0 and 2 x 2  5 x  12  0 , then x is:
A)  12
B)  4
3
C) 
2
2
D) 
3
E) none of these
3. If x  y  2 and x  y  3 , then y 2  x 2 is:
A)  6
B)  1
C) 1
D) 6
E) none of these
4. If 2x  1  3 , then the value of x 2  3 x is:
A)  10
B)  2
C) 2
D) 10
E) none of these
5. The largest value of x such that 3 x divides 10! (with 0 remainder) is:
A) 0
B) 1
C) 2
D) 3
E) none of these
6. Where defined, ( x 1  y 1 )1 is:
xy
A)
yx
B) x  y
1
C)
x y
1
D) 2
x  y2
E) none of these
7. If 3  x  3 x , then x is:
A) 0
3
B)
8
C) 1
D) 6
E) none of these
8. If f ( x)  2 x 2  3 , g ( x)  x  2 , and h( x)  g[ f ( x)] , then h( 2) is:
A)  9
B) 0
C) 3
D) 9
E) none of these
9. The value of
A)
B)
C)
D)
E)
3  2  3 3  6 is:
8 4 3
42 3
42 3
84 3
none of these
10. The sum of all values of x that satisfy 3 
A)
B)
C)
D)
E)
1
1
3
5
none of these
2
 x  0 is:
x
11. Using interval notation, the solution set for  6  3  2x  4 is:
 1 9
A)   , 
 2 2
 1 9
B)  , 
 2 2
 9 1
C)  , 
 2 2
1  9 

D)   ,    ,  
2  2 

E) none of these
12. Using interval notation, the solution set for x 2  x  12   6 is:
A)  2,3
B)  3,2
C) 2, 
D)  ,3
E) none of these
13. Using interval notation, the solution set for 2 x  5  9 is:
A)
B)
C)
D)
E)
 2, 
 ,2  7, 
 ,7
 ,7  2, 
none of these
14. The domain of the real-valued function defined by f ( x)  4 x  2 is:
A) x  0
B) 0  x  2
1
C) x 
2
D) x  2
E) none of these
3x  1 for x  1
15. If f ( x)  
, then  f 0 f 2 is:
2  x for x  1
A)  10
B)  7
C) 0
D) 7
E) none of these
16. If f ( x)  2 x  1 , then, for h  0 ,
A)
B)
C)
D)
E)
2h  1
h
2
1
h
1
2
h
0
none of these
f ( x  h)  f ( x )
is:
h
17. The number of points of intersection for the graphs of x 2  y 2  1 and y 
A)
B)
C)
D)
E)
1
is:
x2
0
1
2
3
none of these
18. A circle with center (1,1) has a diameter of length 4. The coordinates of a point on the
circle are:
A) (1,0)
B) (1,3)
C) (0,1)
D) (2,2)
E) none of these
19. The sum of the digits of the integer represented by 10 95  85 is:
A) 6
B) 834
C) 842
D) 843
E) none of these
20. If the lengths, in inches, of the edges of a rectangular prism are 4, 5 and 6 respectively,
then the surface area, in square inches, of the prism is:
A) 74
B) 120
C) 148
D) 296
E) none of these
21. The number of digits in the integer represented by (25 51 )( 4 54 ) is:
A) 51
B) 54
C) 102
D) 104
E) none of these
22. If an equilateral triangle is inscribed in a unit circle, then the perimeter of this triangle is:
A) 3
B) 3
C) 3 2
D) 3 3
E) none of these
23. If Al and Bob each mentally select one integer at random from 0,1, 2, 3, 4, then the
probability that the absolute value of the difference between Al’s integer and Bob’s
integer is greater than one is:
6
A)
25
9
B)
25
2
C)
5
12
D)
25
E) none of these
24. The sum of the values of m such that x 2  m(2 x  8)  15  0 has equal roots is:
A)  8
B)  2
C) 2
D) 8
E) none of these
25. If x 2  4 x 2  x  1  x , then x is:
A)  1
1
B) 
3
1
C)
3
D) 1
E) none of these
26. If x and y are consecutive odd positive integers such that xy  783 , then x  y is:
A) 56
B) 58
C) 62
D) 64
E) none of these
27. If x 3  2 is divided by x 2  2 , then the remainder is:
A)  2
B) 2
C) 2x  2
D) 2x  2
E) none of these
28. If a  b  3 , b  c  5 , and a  c  4 , then abc is:
A) 6
B) 8
C) 10
D) 12
E) none of these
29. If x 0.3  8 , then x 0.4 equals:
A) 2
32
B)
3
C) 16
D) 96
E) none of these
30. If a  2110 , b  4 56 , and c  8 36 are arranged in decreasing numerical order, then the
ordering is:
A) b  a  c
B) b  c  a
C) a  c  b
D) c  b  a
E) none of these
31. If f (2 x  1)  3x  2 , then f (x) is:
3x  1
A)
2
B) 3x  2
x2
C)
3
D) 2x  3
E) none of these
32. The number of points of intersection of the graphs of x 2  y 2  4 and y  2  x is:
A) 0
B) 1
C) 2
D) 3
E) none of these
33. The number of distinct solutions of 3 x  3  x  2 is:
A) 0
B) 1
C) 2
D) 3
E) none of these
34. If log 10 ( x  8)  log 10 ( x  1)  1 , then x is:
A) 1
B) 2
C) 3
D) 4
E) none of these
35. If sin x  
A)  2
1
and tan x  0 , then sec x is:
2
3
2
3
2
B) 
C)
D) 2
E) none of these
36. If y  f ( x)   x 2  1 for all real numbers x , then the range of f is:
A)  ,1
B) 0,  
C)  ,1
D)  , 
E) none of these
37. The coordinates for the vertex of the graph of y  2 x 2  x  4 are:
 1 31 
A)   , 
 4 8
B) (0,4)
C) (1,6)
 1 
D)   ,4 
 2 
E) none of these

 3 
38. The value of cos  tan 1    is:
 4 

4
A) 
3
4
B) 
5
4
C)
5
5
D)
4
E) none of these
 5
39. If f ( x)  cos( 2 x) , then f 
 6
3
A) 
2
1
B) 
2
1
C)
2
3
D)
2
E) none of these

 is:

40. If n 3  3n 2  2n is divided by 3 for all positive integers n , then the sum of all the
remainders is:
A) 0
B) 1
C) 2
D) 3
E) none of these
41. The smallest distance between a point satisfying x  1   y  1  1 and a point
2
2
satisfying x  4   y  3  4 is:
2
2
A) 13  5
B) 13  4
C) 13  3
D) 13  5
E) none of these
42. The area of the largest square that can be inscribed in a unit circle is:
A) 1
B) 2
C) 4
D) 4 2
E) none of these
43. An urn contains five white balls and three black balls. If three balls are drawn at random
without replacement, then the probability that the three balls drawn are white is:
3
A)
8
3
B)
5
5
C)
8
D) 1
E) none of these
44. The length of the radius of a circle whose area would be doubled by increasing its radius
by 1 is:
A)
3
2
3
2
5
C) 1 
2
D) 1+ 2
E) none of these
B) 1 
45. The lengths of the legs of a right triangle are 5 and 10. If the length of the hypotenuse of
a similar triangle is 15, then the area of the larger triangle is:
A) 25
B) 45
C) 50
D) 60
E) none of these
kx  y  k 2
46. The set of all values of k for which the system 
does not have a solution
x

ky

1

x, y  is:
A)
B)
C)
D)
E)
 1,0
 1,1
0,1
1
none of these
47. If f ( x)  x  2  x  4  2 x  6 for all real numbers, then the maximum value of f is:
A) 2
B) 3
C) 6
D) 20
E) none of these
48. The squares on a chess board are numbered consecutively from 1 to 64. If 3n  2 pebbles
are placed on the n th square, then the total number of pebbles on the chess board is:
A) 6109
B) 6110
C) 6111
D) 6112
E) none of these
49. The number of values of m for which the graphs of x 2  y 2  2 and y 
m
have exactly
x
two points of intersection is:
A)
B)
C)
D)
E)
0
1
2
3
none of these
50. If the vertices of an equilateral triangle are at (0,4), (4,1), and x, y  where x  0 , then x
is:
52 2
A)
3
B) 1
C) 2
43 3
D)
2
E) none of these