DISINI test_07_sol
UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTIETH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 13, 2007
1) Given real numbers x, y and z, define
=
3, 2, – 4
2) Simplify
32 –
3 2 2 –4
–4 3
3 2 2 2 –4 2
32 –
2
32
2
2
=
6 – 8 – 12
9
4
=–
16
xy
x2
32 –
3, 2, – 4 .
14
29
2
32 –
2
32
2
–2
=
2
yz zx
. Evaluate
y2 z2
.
=
32
=
x, y, z
32
2
32 –
2
32
2
2
32
–
2
2
2
2
=
32 – 16
32 – 2
2
=
18
30
=
3
5
3) The midpoints of the longer sides of a 2 by 4
rectangle are joined to the opposite vertices as
indicated in the figure. Find the area of the
shaded quadrilateral.
Area = 2(4) – 2
4)
1
2
1
2 2 –2
2
1 2 =8–4–2=2
Given three consecutive integers, the difference between the cubes of the two largest integers is 666 more than the difference
between the cubes of the two smallest integers. What is the largest integer ?
Let the integers be n – 1, n and n + 1
n
1
3
– n3 = 666 + n3 – (n
–1
3
n3 + 3 n2 + 3n + 1 – n3 = 666 + n3 – n3 + 3 n2 – 3n + 1
6n = 666
5)
n = 111
n + 1 = 112
If w is 10% larger than x, x is 20% larger than y and y is 25% smaller than z, by what percent is w smaller than z ?
w = 1.1x, x = 1.2y and y = .75z
w = 1.1(1.2)(.75)z = .99z
Thus w is 1% smaller that z.
6) A ladder leans against a wall. The top of the ladder is 8 feet above the ground. If the bottom of the ladder is then moved 2 feet
farther from the wall, the top of the ladder will rest against the foot of the wall. How long is the ladder ?
L
8
x
x = L – 2 and L2
L=x+2
L2 = (L – 2
2
x2
82
L2 = L2 – 4L + 4 + 64
+ 64
4L = 68
L = 17
A
7) ABCD is a square, ABE is an equilateral triangle and
B
F is the point of intersection of AC and BE.
Find the degree measure of EAF.
F
E
D
ABE equilateral
EAF =
EAB = 60°. AC a diagonal of the square
EAB –
C
FAB = 45°
FAB = 60° – 45° = 15°
8) Joe has a bag of marbles. Joe gives Larry half of his marbles and two more. Joe then gives Doug half of the marbles he has left
and two more. Finally, Joe gives Jack half of the marbles he has left and two more. Joe has one marble remaining in his bag.
With how many marbles did Joe start ?
Let J be the number of marbles Joe has to start.
1
1
After first gift: J - 2 J – 2 = 2 J – 2
After second gift:
After third gift:
1
J
2
1
J
4
–2–
–3–
1
2
1
2
1
4
1
2
1
J–2 – 2 = 4 J – 3
1
J–3 – 2 = 8 J –
7
2
1
J
8
=1
=
9
2
J=8
9
2
= 36
B
9) The square AFGH is cut out of the rectangle
C
x
2
ABCD, leaving an area of 92 in . If FB = 4 inches
and DH = 8 inches, find the original area of ABCD.
F
G
y
y–4
A
x−8
D
H
y–4=x–8
x = y + 4, xy – (y – 4 2 = 92
(y + 4)(y) – y2 – 8 y
16 = 92
y2 + 4y – y2 + 8y – 16 = 92
12y = 108
y=9
x = 13
Area = 13(9) = 117
10) Suppose that 60 percent of the population has a particular virus. A medical test accurately detects the virus in 90 percent of the
cases in which the patient has the virus, but falsely detects the virus in 20 percent of the cases in which the patient does not have
the virus. If the same patient is tested twice, the results are said to be inconsistent if the results of the two tests do not agree.
If the test is administered twice to all patients, how many patients out of 250 would expect to get inconsistent results ?
Let H = number of the 250 who have the virus. H = (.6)(250) = 150
Let D = number of the 250 who don’t have the virus. D = (.4)(250) = 100
For those who have the virus the number with different results on the two tests is 2(.9)(.1)(150) = 27
For those who don’t have the virus the number with different results on the two tests is 2(.8)(.2)(100) = 32
Note that the factor of two accounts for the order of the results.
Thus, the number with different results on the two tests is 27 + 32 = 59
2
11) Find all real solutions of 9 x + 2 3 x
9x + 2 3x
2
32 x + 2 32 · 3 x = 243
= 243
y2 + 18 y – 243 = 0
Let y = 3 x
y = 3x
= 243.
9 = 3x
y = – 27 , 9
x=2
12) Suppose that sin(2x) =
1
. Express sin 4 x + cos 4 x as a rational number in lowest terms.
7
1
sin(2x) = 2sin x cos x =
1 = sin2 x
(y + 27)(y – 9) = 0
cos2 x
2
= sin4 x
cos4 x = 1 – 2 sin x cos x
sin4 x
cos4 x = 1 – 2
2
1
7
2
7
2 sin2 x cox2 x
sin4 x
2
1
sin(x)cos(x) =
7
=1–
cos4 x
2
1
14
=
13
14
13) For each positive integer k, let a k be the sum of the first k positive integers. If exactly x of the a k consist of one digit,
exactly y of the a k consist of two digits and exactly z of the a k consist of three digits, what is the product x · y · z ?
ak = 1 + 2 + 3 + · · · + k =
k k 1
2
< 10
k k 1
2
< 100
k k 1
2
< 1000
k k 1
2
k(k+1) < 20
k(k+1) < 200
k(k+1) < 2000
x · y · z = 3(10)(31) = 930
k2
k – 20
k2
0
k – 200
k2
(k + 5)(k – 4) < 20
0
k – 2000
k<
0
–1
k<
1
2
–1
800
1 8000
2
k
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTIETH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 13, 2007
1) Given real numbers x, y and z, define
=
3, 2, – 4
2) Simplify
32 –
3 2 2 –4
–4 3
3 2 2 2 –4 2
32 –
2
32
2
2
=
6 – 8 – 12
9
4
=–
16
xy
x2
32 –
3, 2, – 4 .
14
29
2
32 –
2
32
2
–2
=
2
yz zx
. Evaluate
y2 z2
.
=
32
=
x, y, z
32
2
32 –
2
32
2
2
32
–
2
2
2
2
=
32 – 16
32 – 2
2
=
18
30
=
3
5
3) The midpoints of the longer sides of a 2 by 4
rectangle are joined to the opposite vertices as
indicated in the figure. Find the area of the
shaded quadrilateral.
Area = 2(4) – 2
4)
1
2
1
2 2 –2
2
1 2 =8–4–2=2
Given three consecutive integers, the difference between the cubes of the two largest integers is 666 more than the difference
between the cubes of the two smallest integers. What is the largest integer ?
Let the integers be n – 1, n and n + 1
n
1
3
– n3 = 666 + n3 – (n
–1
3
n3 + 3 n2 + 3n + 1 – n3 = 666 + n3 – n3 + 3 n2 – 3n + 1
6n = 666
5)
n = 111
n + 1 = 112
If w is 10% larger than x, x is 20% larger than y and y is 25% smaller than z, by what percent is w smaller than z ?
w = 1.1x, x = 1.2y and y = .75z
w = 1.1(1.2)(.75)z = .99z
Thus w is 1% smaller that z.
6) A ladder leans against a wall. The top of the ladder is 8 feet above the ground. If the bottom of the ladder is then moved 2 feet
farther from the wall, the top of the ladder will rest against the foot of the wall. How long is the ladder ?
L
8
x
x = L – 2 and L2
L=x+2
L2 = (L – 2
2
x2
82
L2 = L2 – 4L + 4 + 64
+ 64
4L = 68
L = 17
A
7) ABCD is a square, ABE is an equilateral triangle and
B
F is the point of intersection of AC and BE.
Find the degree measure of EAF.
F
E
D
ABE equilateral
EAF =
EAB = 60°. AC a diagonal of the square
EAB –
C
FAB = 45°
FAB = 60° – 45° = 15°
8) Joe has a bag of marbles. Joe gives Larry half of his marbles and two more. Joe then gives Doug half of the marbles he has left
and two more. Finally, Joe gives Jack half of the marbles he has left and two more. Joe has one marble remaining in his bag.
With how many marbles did Joe start ?
Let J be the number of marbles Joe has to start.
1
1
After first gift: J - 2 J – 2 = 2 J – 2
After second gift:
After third gift:
1
J
2
1
J
4
–2–
–3–
1
2
1
2
1
4
1
2
1
J–2 – 2 = 4 J – 3
1
J–3 – 2 = 8 J –
7
2
1
J
8
=1
=
9
2
J=8
9
2
= 36
B
9) The square AFGH is cut out of the rectangle
C
x
2
ABCD, leaving an area of 92 in . If FB = 4 inches
and DH = 8 inches, find the original area of ABCD.
F
G
y
y–4
A
x−8
D
H
y–4=x–8
x = y + 4, xy – (y – 4 2 = 92
(y + 4)(y) – y2 – 8 y
16 = 92
y2 + 4y – y2 + 8y – 16 = 92
12y = 108
y=9
x = 13
Area = 13(9) = 117
10) Suppose that 60 percent of the population has a particular virus. A medical test accurately detects the virus in 90 percent of the
cases in which the patient has the virus, but falsely detects the virus in 20 percent of the cases in which the patient does not have
the virus. If the same patient is tested twice, the results are said to be inconsistent if the results of the two tests do not agree.
If the test is administered twice to all patients, how many patients out of 250 would expect to get inconsistent results ?
Let H = number of the 250 who have the virus. H = (.6)(250) = 150
Let D = number of the 250 who don’t have the virus. D = (.4)(250) = 100
For those who have the virus the number with different results on the two tests is 2(.9)(.1)(150) = 27
For those who don’t have the virus the number with different results on the two tests is 2(.8)(.2)(100) = 32
Note that the factor of two accounts for the order of the results.
Thus, the number with different results on the two tests is 27 + 32 = 59
2
11) Find all real solutions of 9 x + 2 3 x
9x + 2 3x
2
32 x + 2 32 · 3 x = 243
= 243
y2 + 18 y – 243 = 0
Let y = 3 x
y = 3x
= 243.
9 = 3x
y = – 27 , 9
x=2
12) Suppose that sin(2x) =
1
. Express sin 4 x + cos 4 x as a rational number in lowest terms.
7
1
sin(2x) = 2sin x cos x =
1 = sin2 x
(y + 27)(y – 9) = 0
cos2 x
2
= sin4 x
cos4 x = 1 – 2 sin x cos x
sin4 x
cos4 x = 1 – 2
2
1
7
2
7
2 sin2 x cox2 x
sin4 x
2
1
sin(x)cos(x) =
7
=1–
cos4 x
2
1
14
=
13
14
13) For each positive integer k, let a k be the sum of the first k positive integers. If exactly x of the a k consist of one digit,
exactly y of the a k consist of two digits and exactly z of the a k consist of three digits, what is the product x · y · z ?
ak = 1 + 2 + 3 + · · · + k =
k k 1
2
< 10
k k 1
2
< 100
k k 1
2
< 1000
k k 1
2
k(k+1) < 20
k(k+1) < 200
k(k+1) < 2000
x · y · z = 3(10)(31) = 930
k2
k – 20
k2
0
k – 200
k2
(k + 5)(k – 4) < 20
0
k – 2000
k<
0
–1
k<
1
2
–1
800
1 8000
2
k