DISINI exam2004
High S
hool Math Contest
University of South Carolina
February 7, 2004
C
1. Given that \ACB is a right angle, what is
the area of 4ABD in the gure shown?
5
6
D
3
A
(a) 6
(b) 7
(
) 8
B
(d) 9
(e) 10
2. Whi
h of the numbers below is a solution to the following equation?
p
(a)
p
6
p
(b) 2 3
1
3
x+
p
3+x=x
p
3 3
(
)
2
p
p
(d) 6 + 2
p
(e) 2 2
D
3. What is the sum of the distan
es AD and
BD in the gure shown?
(b) 28
(
) 29
C
8
A
(a) 27
6
21
(d) 30
B
(e) 31
4. What is the value of the produ
t (log2 3) (log3 5) (log5 8) ?
(a) 2
(b) 3
(
) 4
(d) 5
(e) 6
5. If sin x = 2
os x, then what is the value of sin x
os x ?
(a) 1=3
(b) 2=3
(
) 1=4
1
(d) 1=5
(e) 2=5
6. Suppose that f (x) = ax + b where a and b are real numbers. Given that
f (f (f (x))) = 8x + 21, what is the value of a + b ?
(a) 2
(b) 3
(
) 4
(d) 5
(e) 6
A
7. Suppose AB is a diameter of the
ir
le shown,
Æ
BC is tangent to the
ir
le, \BAC = 30 , and
p
CD =
3. What is the distan
e from A to B ?
D
C
B
p
(a) 3 3
p
(b) 6
(
) 4 3
(d) 8
p
(e) 5 3
8. A box
ontains 4 fair
oins and 6 biased
oins. Whenever a fair
oin is
ipped, it
omes up heads with probability 0:5. Whenever a biased
oin
is
ipped, it
omes up heads with probability 0:8. A
oin is randomly
hosen from the box and then
ipped. What is the probability that it
will
ome up heads?
(a) 0:6
9. The expression
(a) 3
(b) 0:64
p
(
) 0:68
log2 9
2
(b) 4:5
(d) 0:72
(e) 0:76
(d) 7:5
(e) 9
simpli es to
(
) 6
10. If a + b = 2 and a2 + b2 = 5, then what is the value of a3 + b3 ?
(a) 11
(b) 12
(
) 13
(d) 14
(e) 15
11. How many di erent real roots does the following polynomial have?
x
(a) 0
(b) 1
11
+ x10 + x9 + + x + 1
(
) 3
2
(d) 5
(e) 7
12. What is the ratio of the area of a square that
ir
ums
ribes a
ir
le to
the area of a square that ins
ribes the same
ir
le?
(a)
p
2
(b)
p
3
13. Suppose that a, b,
, d, and
three equations below.
(
) 2
e
(d)
p
6
(e) 3
are numbers whi
h satisfy the system of
8 13
<
: 65
+ 26b + 2
+ 13d + 3e = 18
a + 12b +
+ 6d + e = 7
a + 10b +
+ 5d + e = 6
a
What is the value of e ?
(a) 1
(b) 0
(
) 1
(d) 2
(e) 3
1
14. Suppose x is a
omplex number for whi
h x + = 2
os 12Æ. What is the
x
1
5
value of x + 5 ?
x
(a) 4
(b) 3
(
) 2
(d) 1
(e) 0
15. What is the sum of all the di erent solutions to the following equation?
(x2 + 1)(x4 + 1)(x6 + 1)
+x
x + 1
(a) 4
(b) 3
(
) 2
1=0
(d) 1
(e) 0
16. For how many numbers n 2 f1; 2; 3; : : : ; 2004g is the quantity 4n6 + n3 +5
a multiple of 7 ?
(a) 0
(b) 285
(
) 286
(d) 287
(e) 2004
17. How many sequen
es a1; a2; a3 ; a4; a5 satisfying a1 < a2 < a3 < a4 < a5
an be formed if ea
h ai must be
hosen from the set f1; 2; 3; 4; 5; 6; 7; 8; 9g ?
(a) 118
(b) 120
(
) 122
3
(d) 124
(e) 126
18. How many di erent real numbered pairs (x; y ) satisfy the system of two
equations below?
(a) 6
x + xy + y = 9
x2 + y 2 = 17
(b) 4
(
) 3
(d) 2
(e) 0
19. Suppose that a is a non-zero real number for whi
h sin x + sin y = a and
os x +
os y = 2a. What is the value of
os (x y ) ?
(a)
a2
2
2
(b)
3a2 2
2
(
)
5a2 2
2
(d)
7a2 2
2
(e)
9a2 2
2
20. Seven people are sitting in a theater wat
hing a show. The row they are
in
ontains seven seats. After intermission, they return to the same row
but
hoose seats randomly. What is the probability that neither of the
people sitting in the two aisle seats was previously sitting in an aisle seat?
(a) 3=7
(b) 10=21
(
) 11=21
(d) 4=7
21. The six points A; B; C; D; E; F are equally spa
ed
along the
ir
umferen
e of a
ir
le of radius 1.
Line segments AC , CE , AE , BD, DF , and BF
are drawn, and the resulting star-shaped region is
shaded in. What is the area of this star-shaped
region?
p
2 6
(a)
3
(b)
p
3
2
(
)
3
4
(e) 13=21
E
F
D
A
C
p
3 2
(d)
2
B
(e)
3
4
22. From a
lass of 12 students, 3 are
hosen to form a math
ontest team.
The team is required to in
lude at least one boy and at least one girl.
If exa
tly 160 di erent teams
an be formed from the 12 students, then
whi
h of the following
an be the di eren
e between the number of boys
and the number of girls in the
lass?
(a) 0
(b) 2
(
) 4
(d) 6
23. In the gure shown, \ABC and \BDA are
both right angles. If v + w = 35 and x + y = 37,
then what is the value of y ?
B
w
C
(a) 11
(b) 12
(
) 13
(e) 8
v
y
D
A
x
(d) 14
(e) 15
24. Suppose is an angle between 0Æ and 90Æ for whi
h
os ()
os (2) = 1=4.
What is the value of ?
(a) 32Æ
(b) 34Æ
(
) 36Æ
(d) 38Æ
(e) 40Æ
25. How many integers do the following nite arithmeti
progressions have in
ommon?
1; 8; 15; 22; : : : ; 2003
(a) 10
(b) 23
and
(
) 25
2; 13; 24; 35; : : : ; 2004
(d) 26
(e) 27
26. A standard, fair, 6-sided die is rolled 8 times. Given that the number
3 appears exa
tly three times, what is the probability that no two 3's
appear on
onse
utive rolls?
(a) 5=14
(b) 3=7
(
) 1=2
5
(d) 4=7
(e) 9=14
27. A point (x; y ) is
alled integral if both
x
and
y
are integers. How many
integral points are inside of the parallelogram whose four sides are on the
lines
x
= 100,
(a) 73
28. Let
O
x
= 300,
y
1
= 3 x + 0:1, and
(b) 71
ABC D
y
(
) 69
(d) 67
BD .
The areas of
4AOB
and
AC
4C OD
ABC D
C
Area = 4
O
are
respe
tively 9 and 4. What is the minimum
possible area of the quadrilateral
(e) 65
D
be the quadrilateral shown, where
is the interse
tion of the line segments
and
1
= 3 x + 0:6 ?
?
Area = 9
A
B
(a) 22
(b) 23
(
) 24
2003 digits
z
}|
(d) 25
(e) 26
2003 digits
{
z
}|
{
29. Let a be the integer 333 333 and let b be the integer 666 666. What is
the 2004th digit (
ounting from the right) that appears in the produ
t
(a) 0
(b) 1
(
) 2
(d) 7
ab
?
(e) 8
30. In how many ways
an 2004 be written as a sum of two or more
onse
utive
positive integers written in in
reasing order? For example, we
onsider
667 + 668 + 669 as an a
eptable sum but not 669 + 668 + 667.
(a) 1
(b) 2
(
) 3
6
(d) 4
(e) 5
hool Math Contest
University of South Carolina
February 7, 2004
C
1. Given that \ACB is a right angle, what is
the area of 4ABD in the gure shown?
5
6
D
3
A
(a) 6
(b) 7
(
) 8
B
(d) 9
(e) 10
2. Whi
h of the numbers below is a solution to the following equation?
p
(a)
p
6
p
(b) 2 3
1
3
x+
p
3+x=x
p
3 3
(
)
2
p
p
(d) 6 + 2
p
(e) 2 2
D
3. What is the sum of the distan
es AD and
BD in the gure shown?
(b) 28
(
) 29
C
8
A
(a) 27
6
21
(d) 30
B
(e) 31
4. What is the value of the produ
t (log2 3) (log3 5) (log5 8) ?
(a) 2
(b) 3
(
) 4
(d) 5
(e) 6
5. If sin x = 2
os x, then what is the value of sin x
os x ?
(a) 1=3
(b) 2=3
(
) 1=4
1
(d) 1=5
(e) 2=5
6. Suppose that f (x) = ax + b where a and b are real numbers. Given that
f (f (f (x))) = 8x + 21, what is the value of a + b ?
(a) 2
(b) 3
(
) 4
(d) 5
(e) 6
A
7. Suppose AB is a diameter of the
ir
le shown,
Æ
BC is tangent to the
ir
le, \BAC = 30 , and
p
CD =
3. What is the distan
e from A to B ?
D
C
B
p
(a) 3 3
p
(b) 6
(
) 4 3
(d) 8
p
(e) 5 3
8. A box
ontains 4 fair
oins and 6 biased
oins. Whenever a fair
oin is
ipped, it
omes up heads with probability 0:5. Whenever a biased
oin
is
ipped, it
omes up heads with probability 0:8. A
oin is randomly
hosen from the box and then
ipped. What is the probability that it
will
ome up heads?
(a) 0:6
9. The expression
(a) 3
(b) 0:64
p
(
) 0:68
log2 9
2
(b) 4:5
(d) 0:72
(e) 0:76
(d) 7:5
(e) 9
simpli es to
(
) 6
10. If a + b = 2 and a2 + b2 = 5, then what is the value of a3 + b3 ?
(a) 11
(b) 12
(
) 13
(d) 14
(e) 15
11. How many di erent real roots does the following polynomial have?
x
(a) 0
(b) 1
11
+ x10 + x9 + + x + 1
(
) 3
2
(d) 5
(e) 7
12. What is the ratio of the area of a square that
ir
ums
ribes a
ir
le to
the area of a square that ins
ribes the same
ir
le?
(a)
p
2
(b)
p
3
13. Suppose that a, b,
, d, and
three equations below.
(
) 2
e
(d)
p
6
(e) 3
are numbers whi
h satisfy the system of
8 13
<
: 65
+ 26b + 2
+ 13d + 3e = 18
a + 12b +
+ 6d + e = 7
a + 10b +
+ 5d + e = 6
a
What is the value of e ?
(a) 1
(b) 0
(
) 1
(d) 2
(e) 3
1
14. Suppose x is a
omplex number for whi
h x + = 2
os 12Æ. What is the
x
1
5
value of x + 5 ?
x
(a) 4
(b) 3
(
) 2
(d) 1
(e) 0
15. What is the sum of all the di erent solutions to the following equation?
(x2 + 1)(x4 + 1)(x6 + 1)
+x
x + 1
(a) 4
(b) 3
(
) 2
1=0
(d) 1
(e) 0
16. For how many numbers n 2 f1; 2; 3; : : : ; 2004g is the quantity 4n6 + n3 +5
a multiple of 7 ?
(a) 0
(b) 285
(
) 286
(d) 287
(e) 2004
17. How many sequen
es a1; a2; a3 ; a4; a5 satisfying a1 < a2 < a3 < a4 < a5
an be formed if ea
h ai must be
hosen from the set f1; 2; 3; 4; 5; 6; 7; 8; 9g ?
(a) 118
(b) 120
(
) 122
3
(d) 124
(e) 126
18. How many di erent real numbered pairs (x; y ) satisfy the system of two
equations below?
(a) 6
x + xy + y = 9
x2 + y 2 = 17
(b) 4
(
) 3
(d) 2
(e) 0
19. Suppose that a is a non-zero real number for whi
h sin x + sin y = a and
os x +
os y = 2a. What is the value of
os (x y ) ?
(a)
a2
2
2
(b)
3a2 2
2
(
)
5a2 2
2
(d)
7a2 2
2
(e)
9a2 2
2
20. Seven people are sitting in a theater wat
hing a show. The row they are
in
ontains seven seats. After intermission, they return to the same row
but
hoose seats randomly. What is the probability that neither of the
people sitting in the two aisle seats was previously sitting in an aisle seat?
(a) 3=7
(b) 10=21
(
) 11=21
(d) 4=7
21. The six points A; B; C; D; E; F are equally spa
ed
along the
ir
umferen
e of a
ir
le of radius 1.
Line segments AC , CE , AE , BD, DF , and BF
are drawn, and the resulting star-shaped region is
shaded in. What is the area of this star-shaped
region?
p
2 6
(a)
3
(b)
p
3
2
(
)
3
4
(e) 13=21
E
F
D
A
C
p
3 2
(d)
2
B
(e)
3
4
22. From a
lass of 12 students, 3 are
hosen to form a math
ontest team.
The team is required to in
lude at least one boy and at least one girl.
If exa
tly 160 di erent teams
an be formed from the 12 students, then
whi
h of the following
an be the di eren
e between the number of boys
and the number of girls in the
lass?
(a) 0
(b) 2
(
) 4
(d) 6
23. In the gure shown, \ABC and \BDA are
both right angles. If v + w = 35 and x + y = 37,
then what is the value of y ?
B
w
C
(a) 11
(b) 12
(
) 13
(e) 8
v
y
D
A
x
(d) 14
(e) 15
24. Suppose is an angle between 0Æ and 90Æ for whi
h
os ()
os (2) = 1=4.
What is the value of ?
(a) 32Æ
(b) 34Æ
(
) 36Æ
(d) 38Æ
(e) 40Æ
25. How many integers do the following nite arithmeti
progressions have in
ommon?
1; 8; 15; 22; : : : ; 2003
(a) 10
(b) 23
and
(
) 25
2; 13; 24; 35; : : : ; 2004
(d) 26
(e) 27
26. A standard, fair, 6-sided die is rolled 8 times. Given that the number
3 appears exa
tly three times, what is the probability that no two 3's
appear on
onse
utive rolls?
(a) 5=14
(b) 3=7
(
) 1=2
5
(d) 4=7
(e) 9=14
27. A point (x; y ) is
alled integral if both
x
and
y
are integers. How many
integral points are inside of the parallelogram whose four sides are on the
lines
x
= 100,
(a) 73
28. Let
O
x
= 300,
y
1
= 3 x + 0:1, and
(b) 71
ABC D
y
(
) 69
(d) 67
BD .
The areas of
4AOB
and
AC
4C OD
ABC D
C
Area = 4
O
are
respe
tively 9 and 4. What is the minimum
possible area of the quadrilateral
(e) 65
D
be the quadrilateral shown, where
is the interse
tion of the line segments
and
1
= 3 x + 0:6 ?
?
Area = 9
A
B
(a) 22
(b) 23
(
) 24
2003 digits
z
}|
(d) 25
(e) 26
2003 digits
{
z
}|
{
29. Let a be the integer 333 333 and let b be the integer 666 666. What is
the 2004th digit (
ounting from the right) that appears in the produ
t
(a) 0
(b) 1
(
) 2
(d) 7
ab
?
(e) 8
30. In how many ways
an 2004 be written as a sum of two or more
onse
utive
positive integers written in in
reasing order? For example, we
onsider
667 + 668 + 669 as an a
eptable sum but not 669 + 668 + 667.
(a) 1
(b) 2
(
) 3
6
(d) 4
(e) 5