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Transcript
Junior Olympiad
1.
The sum of the solutions to 32 x  9 3 x 4 is:
2
A)
B)
C)
D)
E)
2.
The radius of the circle 3x 2  3 y 2  30 x  18 y  90  0 is:
A)
B)
C)
D)
E)
3.
3
6
9
36
none of these
A quadratic equation whose roots are the reciprocals of the roots of 6 x 2  17 x  14  0 is:
A)
B)
C)
D)
E)
5.
1
2
3
4
none of these
If a sphere has a volume of 36 cubic feet, then the length of the radius of the sphere, in feet, is:
A)
B)
C)
D)
E)
4.
0
3
5
9
none of these
1 2 1
1
x  x
0
6
17
14
 6 x 2  17 x  14  0
14 x 2  17 x  6  0
6 17 1
 
0
x 2 x 14
none of these
If 2 x 3  5 y 2  61 and 3x 3  2 y 2  42 , then xy equals:
A)
B)
C)
D)
E)
1
2
4
6
none of these
6.
The value of k such that the vertex of y  5 x 2  7 x  k is on the x-axis is:
A) 0
20
B)
49
C) 1
49
D)
20
E) none of these
7.
If a set of four parallel lines is intersected by a set of six parallel lines not parallel to the lines in
the first set, then the number of parallelograms formed is:
A)
B)
C)
D)
E)
8.
The value of k such that 5 x 2  kx  7  0 has roots whose sum is equal to 5 is:
A)
B)
C)
D)
E)
9.
6
24
90
4!6!
none of these
 25
 10
5
25
none of these
A palindromic integer is an integer that remains unchanged when you reverse the order of its
digits. For example, 123454321 and  131 are palindromic integers. The number of
non-negative palindromic integers less than 1000 is:
A)
B)
C)
D)
E)
90
99
108
109
none of these
10. The smallest integer divisible by 75 that is greater than zero and is written using only 0’s
and 1’s (base 10) is:
A)
B)
C)
D)
E)
100100
111000
1010100
10001100
none of these
11. A three-digit integer is formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9. If no digit is used more
than once, then the number of integers that begin with an even digit and end with an odd digit is:
A)
B)
C)
D)
E)
35
140
210
729
none of these
12. If 3x 4  5 x 3  2 x 2  6 is divided by x + 3, then the sum of the coefficients of the quotient and
the remainder is:
A)
B)
C)
D)
E)
0
25
75
96
none of these
13. If x  2 x  42  3 , then the sum of the solution(s) is:
A)
B)
C)
D)
E)
3
8
11
14
none of these
14. If a, b, c, and d are four distinct positive even integers, then their greatest common factor is:
A)
B)
C)
D)
E)
odd
even
sometimes odd and sometimes even, depending on the values of a, b, c, and d
2
none of these
15. If the sum of two numbers is 14 and the absolute value of their difference is 8, then the absolute
value of the difference of the squares of the two numbers is:
A)
B)
C)
D)
E)
36
64
112
132
none of these
16. An instructor gave a test consisting of 40 multiple-choice questions. She considered two options
for converting the students’ score to a 100 point scale. The first option was to use the percentage
100  raw score
correct; in other words, converted score =
. The second option was to double the
40
students raw score and add 20; in other words, converted score = 2  raw score  20 . One student
remarked that she did not care which option the instructor used, since her converted score would
be the same under both options. The raw score for this student is:
A)
B)
C)
D)
E)
0
25
30
35
none of these
17. If a third of a number subtracted from the sum of three times that number and three and a third is
54, then the number is:
A)
B)
C)
D)
E)
15
21
29
32
none of these
18. If x  0 , then for all h,
A)
B)
C)
D)
E)
3x 3h  2 x 2 h
equals:
xh
5
5x 4h
3x 3  2 x 2
x h 3x h  2
none of these


19. The digits of a 4-digit integer sum to 20. The fourth digit is three times the third digit. The
second digit is one more than twice the third digit, which is one more than twice the first digit.
The product of the four digits is:
A)
B)
C)
D)
E)
105
144
189
945
none of these
20. A 20-foot by 30-foot rectangular barn sits in the middle of a flat, open field. The farmer wants to
tether a goat to the barn using a chain 50 feet long. The goat cannot go under, into, or through the
barn. If the farmer wishes to provide the goat with the maximum possible grazing area, then the
farmer should attach the chain to the barn:
A)
B)
C)
D)
E)
at one of the corners.
in the middle of one of the 20-foot sides.
in the middle of one of the 30-foot sides.
wherever he wants, since all locations provide the same grazing area.
none of these
21. The value of c such that 2 x 2  9 x  c  0 has roots whose product is equal to 7 is:
A)  14
7
B)
2
C) 14
D) 28
E) none of these
22. If 2 x  2 y  8 and 5 x  3 y  9 , then y  x equals:
A) 1
7
B)
4
13
C)
4
D) 4
E) none of these
23. The area of the equilateral triangle that is circumscribed about a circle whose diameter has
length 8 is:
A)
B)
C)
D)
E)
24 3
48 3
96 3
192 3
none of these
24. The distance from the point (1,  2) to the line 3 y  4 x  15  0 is:
A)
5
B)
7
C) 5
25
D)
4
E) none of these
25. If 0  x   , then the sum of the solution(s) of sin x  sin( 2 x)  0 is:
A)
B)
C)
D)
E)

3
2
3
5
6
5
3
none of these
26. An equation of the line that passes through the points (3,1) and (5,  3) is:
A)
B)
C)
D)
E)
1
5
y  x
2
2
1
1
y  x
4
4
y  2 x  5
1
1
y  x
2
2
none of these
27. An equation relating x and y such that any point (x, y) that satisfies the equation is twice as far
from (0, 3) as it is from (0, 0) is:
A)
B)
C)
D)
E)
y 1
x 2  ( y  1) 2  4
( y  2) 2  x 2  1
x 2  ( y  4) 2  4
none of these
28. If cos x 
3
and sin x  0 , then tan x equals:
5
3
4
3

5
3
5
4
3
none of these
A) 
B)
C)
D)
E)
5  x 3 ,

29. If f ( x)   3
 2 x,

 x 1
A)
B)
C)
D)
E)
x 1
x 1
, then the sum of f (2),
f (1), and f (3) is:
57
2
29
31
65
2
none of these
30. If f ( x)  x  3 , then the sum of the x-intercepts of the graph of y  
A)
B)
C)
D)
E)
1
f ( x  7)  2 is:
3
8
6
0
8
none of these
31. Doug’s sprayer has a 300-gallon capacity, and has been mistakenly filled to capacity with a
mixture of chemical and water, of which 25% is chemical. The correct mixture is 20% chemical.
To get the correct mixture Doug needs to drain out some of the old mixture and refill the sprayer
to capacity with water. The amount, in gallons, of the old mixture that must be drained out of the
sprayer is:
A)
B)
C)
D)
E)
15
60
75
220
none of these
32. The solution set of 0  5x  6  7 is:
 6 1
A)   , 
 5 5
 1 6
B)  , 
 5 5
1 6 
C)  , 
5 5 
 6 13 
D)  , 
5 5 
E) none of these
33. If f ( x) 
A)
B)
C)
D)
E)
3
f ( 4  h )  f ( 4)
for x  2 , then for h  0 ,
equals:
x2
h
 3h 2
2(2  h)
3(1  h)
h( 2  h)
 3(2  h)
2h( 4  h)
3
2( 2  h )
none of these
34. If y > 0 and the distance between (5, 6) and (2,  y ) is
A)
B)
C)
D)
E)
218 , then y equals:
5
6
7
19
none of these
35. If a rectangle is bounded by the x-axis and x 2  y 2  36 for y  0 , then the area of the rectangle
as a function of x is:
A) 2 x 36  x 2
B) 2 x(6  x)
C) x 36  x 2
D) x(6  x)
E) none of these
36. If a tractor pulling a 22 foot disc covers m acres per hour and uses 8 gallons of fuel per hour,
then the amount of fuel, in gallons, the tractor uses covering ten acres of ground is:
A)
B)
C)
D)
E)
8
m
m
8
80
m
10m
8
none of these
37. If x  0 , then the solution set of 2  1 
3
x  7 is:
4
4   32 

A)  8,     4, 
3  3 

4 
B)  , 8
3 
 4 32 
C)  , 
3 3 
 32 
D)  4, 
 3
E) none of these
38. If the x-intercepts of the graph of y  x 2  (a  b) x  ab are (2, 0) and (5, 0), then a  b is:
A)
B)
C)
D)
E)
7
3
3
7
none of these


2


39. If x  0 , then the sum of the solution(s) of 3 3x 2  4  11 3x 2  4  20 is:
A) 0
2 2
 3
B)
3
C) 2 2  3
D) 5
E) none of these
40. The sum of the y coordinates of the points of intersection of the graphs of y  x  1 and
x 2  y 2  16 is:
A)
B)
C)
D)
E)
1
0
1
31
none of these
41. If f ( x)  4  3 sin x for all real numbers x, then the range of f is:
A)
B)
C)
D)
E)
[4, 4]
[3, 3]
[7,  1]
[1, 7 ]
none of these
42. If three fair coins are tossed, then the probability of obtaining exactly one tail is:
A)
B)
C)
D)
E)
1
3
3
8
1
2
2
3
none of these
43. Where defined, the inverse of f ( x) 
A)
B)
C)
D)
E)
 2x
x3
x3
2x
x2
3x
3x
x2
none of these
2x
is:
x3
44. If x  y  3
A)
B)
C)
D)
E)
21
4
33
4
57
4
57
2
none of these
45. If f ( x) 
A)
x y
for all real numbers x and y, then  5  2 1 equals:
2
1
1 x
where defined, and f (a)  2 , then f (1  a ) equals:
1
4
3
2
3
C)
4
2 3
D)
3
E) none of these
B)
46. The water in the Flint River flows at 5 kilometers per hour. If a speedboat can go 15 kilometers
upstream in the same time it takes to go 25 kilometers downstream, then the speed of the boat, in
kilometers per hour, in still water is:
A)
B)
C)
D)
E)
10
15
25
30
none of these
0.00052  10 5000  10 
0.0026  10 100000  10 
4
47. When simplified,
A)
B)
C)
D)
E)
10 8
10 9
1010
1011
none of these
8
7
15
equals:
48. The sum of the solution(s) of log( x  1)  log 6  log( x  2)  log x is:
A)
B)
C)
D)
E)
7
3
7
10
none of these
49. The number of ways in which five people can stand in a line if two people refuse to stand side by
side is:
A)
B)
C)
D)
E)
48
72
120
168
none of these
50. If the 27-inch diameter tires of a bicycle are turning at the rate of 125 revolutions per minute, then
the linear velocity of the bicycle in feet per second is:
A)
B)
C)
D)
E)
75
32
75
16
25
3
75
8
none of these