Junior Olympiad
2012
1. If the graph of f ( x)  ax 2  5ax  6 contains the point  4,  10, then a is:
A)  4
5
B) 
2
C) 4
D) 5
E) None of these
2. The length of the diameter of the circle given by x 2  y 2  10 y  4( x  1) is:
A)
B)
C)
D)
E)
5
6
10
20
None of these
3. The sum of all the solutions of sin( 2 x)  cos( 2 x)  0 in the interval 0  x   is:
A)
B)
C)
D)
E)
3
4
5
4
5
2
9
2
None of these
4. If x and y satisfy 2 x  y  5 and x  2 y  5 , then y  x equals:
A)
B)
C)
D)
E)
5
1
1
5
None of these
5. If the product of the roots of 2 x 2  10 x  c  0 is  10, then c is equal to:
A)  20
16
B) 
3
29
C)
3
D) 20
E) None of these
6. If i   1 , then i 485 equals:
A) i
B)  1
C)  i
D) 1
E) None of these
7. Where defined, if
2 3 2
  , then c equals:
a b c
b  3a
ab
2ab
B)
3a  2b
C) a  b
2 ab
D)
3a  b
E) None of these
A)
8. If f ( x)  4  2 sin x for all real numbers x , then the range of f is:
A)
B)
C)
 6, 6
 4, 4
 6,  2
 2, 6
D)
E) None of these
9. The units digit of the product of all the prime numbers less than 700 is:
A) 0
B) 2
C) 5
D) 6
E) None of these
10. The area of the polygon with consecutive vertices 2,  3 , 3, 3 ,  4, 3 and  3,  3 is:
A) 20
B) 25
C) 30
D) 36
E) None of these
11. The value of a  0 such that the graph of y  log a ( x) contains the point 17,13 is:
A)
13
B)
17
17
13
13
C)
17
17
D)
13
E) None of these
12. If f ( x)  4 x 2  7 x  48 , and h  0, then
A)
B)
C)
D)
E)
f ( x  h)  f ( x )
is:
h
8x  7
8x  3h  7
8x  4h  7
8 xh  4h 2  7h
None of these
13. The number of solutions of ln( 2 x  1)  ln( 2 x  3)  ln( x  2)  ln( x  1) is:
A) 0
B) 1
C) 2
D) 3
E) None of these
14. Jones is the chairman of a committee. The number of ways the committee of 5 can be chosen from 10
people, given that Jones must be on the committee, is:
A) 126
B) 252
C) 495
D) 3024
E) None of these
15. If the roots of f ( x)  x 3  4 x 2  8 x  11 are a ,b, and c, then the value of
1
1
1
is:


ab bc ac
8
11
4

11
4
11
8
11
None of these
A) 
B)
C)
D)
E)
2  x 2 for x  1

16. If f ( x)   1
, then the sum of f (3), f (1) and f (3) is:
for x  1

 x 1
7
A) 
4
1
B)
4
3
C)
4
9
D)
4
E) None of these
17. If m and n are the zeros of the equation ax 2  bx  c  0, then the value of
A)
B)
C)
D)
E)
ab  b 2 c
2b 2 c
3ac  b 2
a 3c
3abc  b 3
a 2c
3bc  a 3
b2c
None of these
m2 n2

is:
n
m
18. Derek must choose a four-digit PIN number. If each digit can be chosen from 0 to 9 inclusively, then
the number of all possible PIN numbers that can Derek choose is:
A) 5,040
B) 6,561
C) 9,000
D) 10,000
E) None of these
19. The sum of the prime factors of 1155 is:
A) 18
B) 20
C) 21
D) 22
E) None of these
20. The sum of the values of k for which x 2  2(k  4) x  2k  0 has equal roots is:
A)
B)
C)
D)
E)
10
18
20
24
None of these

21. If xy  0 , then x  y  x 1  y 1

1
is:
A) xy
B)
1
xy
C)
 x  y 2
D) 1
E) None of these
22. If the graph of f ( x)  x 3  3x 2 is translated by 2 units up and 1 unit to the right, then an equation of the
new graph is:
A) h( x)  x 3  6 x 2  9 x  2
B) h( x)  x 3  3x
C) h( x)  x 3  6 x 2  9 x  6
D) h( x)  x 3  3x  4
E) None of these
23. If all the corners of a cube with sides 10 cm are cut off at the midpoints of three adjacent sides, then the
total surface area in cm2 , is:
A) 300

B) 300 2  3

C) 75 4  3




D) 100 3  3
E) None of these
24. Determine k such that x 3  7 x 2  5 x  k  2 is divisible by ( x  3) .
A)
B)
C)
D)
E)
 77
 75
1
77
None of these
25. Using interval notation, the solution set of
2 4
 x
3 5  0 is:
6
2
x
5

A)   , 
6

5

B)  ,  
6


C)
 ,  3  0,
5
6 

5 
D)  3, 0    ,  
6 
E) None of these
26. If point P is on the graph of y  3 x and P minimizes the sum of the squares of the distances from P to
the points (0, 0) , (6, 0) and (3,1 0) , then the x coordinate of P is:
A)
B)
C)
D)
E)
2
5
2
3
7
10
17
5
None of these
27. If a fair nickel is to be flipped at random 6 times, then the probability that it will land on tails more
often than heads is:
7
A)
32
1
B)
3
11
C)
32
3
D)
8
E) None of these
28. The sum of the digits of the integers from 1 to 100, inclusively, is:
A) 801
B) 900
C) 901
D) 5050
E) None of these
29. If an equilateral triangle is rotated 60 degrees with respect to its center, a star-shaped figure is obtained.
If the overlapping hexagon shape in the middle has an area of 90, then the area of the original triangle is:
A) 110
B) 135
C) 150
D) 180
E) None of these
30. In a group of students, everybody received one valentine from each of the other students. If the girls
received a total of 96 valentines and the boys received a total of 176 valentines, then the number of
girls in the group is:
A) 6
B) 11
C) 16
D) 17
E) None of these
2
31. A ball is dropped from a height of 6 feet. If, on each bounce, the ball returns to of its previous height,
3
then the distance the ball travels up and down, in feet, is:
A) 12
B) 18
C) 30
D) 36
E) None of these
32. A chessboard has a square shape subdivided into 64 congruent squares in 8 rows and 8 columns. The
total number of squares on the chessboard is:
A) 65
B) 101
C) 203
D) 204
E) None of these
33. The sum of the solutions of
x  20122  2x  20122
 2011 is:
A) 671
B) 2011
C) 2682
14081
D)
3
E) None of these
34. If a, b, and c are positive integers, such that a log 96 (3)  b log 96 (2)  c , then
ab
is:
c
A 1
B) log 2 (5)  1
C)
log 96 (3)
2  log 96 (2)
D) 6
E) None of these
35. If x  y  1 and x 2  y 2  2 , then x 4  y 4 is:
A)
B)
C)
D)
E)
5
2
3
7
2
4
None of these
36. The four angles of a quadrilateral form an arithmetic sequence. If the second smallest angle is 15
degrees less than twice the smallest angle, then the measure, in degrees, of the largest angle is:
A) 45
B) 111
C) 135
D) 195
E) None of these
37. If a and b are integers such that x 2  x  1 is a factor of ax 3  bx 2  1 , then b is:
A)  2
B)  1
C) 0
D) 1
E) None of these
38. If the total edge length of a rectangular solid is 72, and the total surface area is 214, then the square of
the length of a diagonal of the solid is:
A) 110
B) 217
C) 253
D) 324
E) None of these
39. Marton and his dog walk home from the park. It takes Marton 36 minutes and his dog walks twice as
fast. They start together, but the dog reaches home before Marton and returns to meet with him. After
meeting Marton, the dog walks home, again at double speed, and then turns back to meet Marton again.
After leaving the park, the number of minutes before he meets the dog for the second time is:
A) 24
B) 27
C) 30
D) 34
E) None of these
1
1
40. The solution for 27 6  x 3 
A)
B)
C)
D)
E)
3
22
5 3
is:
5
5
25
125
None of these
41. For a cubic polynomial f (x) , if f 2  16 and f  2  f 0  f 3  0 , then f  1 is:
A)  12
B)  8
C) 64
D) 96
E) None of these
1
42. If sin( 2 x) 
A)
5
, then sin 4 x  cos 4 x is:
2
5
B)
C)
5 1
5
5 1
5
9
10
E) None of these
D)
43. If g ( x)  1  x 2 and f g x  
1 x2
1
for x  0 , then f   is:
2
x
2
3
4
B) 1
A)
C)
2
D) 3
E) None of these
44. For a 3 x 3 grid system with 9 points, if 3 points are selected simultaneously and randomly from the 9
points, then the probability that the points are collinear is:
2
A)
21
3
B)
11
1
C)
7
1
D)
6
E) None of these
45. On every birthday, Anna had a cake that contained her age in candles. She always managed to blow out
all the candles. If she has blown out 325 candles so far, then the number of candles her mom needs to
buy the next three years is:
A) 80
B) 81
C) 101
D) 123
E) None of these
46. If f x  ln 2  ln x , then the domain of f (x) is:
A)
0, e
 1

B)  2 ,  
e

C) 0,  
D)  2, 
E) None of these
47. If f x   3  e 2 x , then f
A)
e
x2
1
x  is:
3
2
B) ln 3  x
C)

1
ln 3  x 2
2

D) ln x 2  3
E) None of these
48. A possible rational function that has zeros at 
and has a horizontal asymptote y 
A)
4 x 2  13 x  10
9 x 2  15 x  4
B)
4 x 2  7 x  15
3x 2  5 x  2
4

 4 x  x  2
3

C)  3x  15 x  3
4

5

3 x  x  3
4

D) 4 x  1 x  2
3

E) None of these
4
is:
3
5
1
and 3, has vertical asymptotes x 
and x  2 ,
3
4
For problems 49 and 50 use the given graph:
10
g
8
6
4
2
f
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
f
49. For the graph above,  f  g 1  g  f 0   1 is:
g
110
A) 
3
14
B) 
3
4
C) 
3
D) Does not exist
E) None of these
g
50. For the graph above, the domain for   x  is:
f
A)  4, 5
B)
C)
 4, 5
 6,  4   4, 5  5, 6
 6, 6
D)
E) None of these