DISINI test_11
UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-FOURTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 9, 2011
1) Evaluate 1
1
10
1
11
1
1
1
12
1
2011 2012
2011 2010
2011
2) Simplify the expression
1
13
. Express your answer as a rational number in lowest terms.
. Express your answer as a rational number in lowest terms.
3) The difference of two positive numbers is 4 and the product of the two numbers is 19. Find the sum of the
two numbers.
4) Find the value of a + b
2
if ab
– 14 and a2
b2
30 .
5 Consider the circle with diameter AD. If ABC 130 °, CDA
and BCA 20 °, find BAD. Express your answer in degrees.
50 °
C
B
A
D
6) If the length of a rectangle is increased by 40% and the width is decreased by 15%, what is the percentage change in the area of
the rectangle?
7) One day last month, Ray's Reasonably Reliable Repair Service offered the following Saturday special:
"Buy 3 shock absorbers at the regular price and receive an 80% discount on the fourth."
Jerome bought 4 shock absorbers on that day and paid a total of $176. What was the regular price of one shock absorber?
8) The two roots of the quadratic equation x2
85 x
9) For real numbers x, y and z, define F x, y, z
xy
c
0 are prime numbers. What is the value of c?
yz
z x. For which real numbers a is F 2, a, a
1
F 5, a, a
1?
10) Michelle has a collection of marbles, all of which are either blue or green. She is creating pairs of 1 blue marble and 1 green marble.
After a while, she notices that
2
3
of all the blue marbles are paired with
3
5
of all the green marbles. What fraction of Michelle’s
marble collection has been paired up? Express your answer as a rational number in lowest terms.
11) Find the integer value of the expression log 7
1
1
8
log 8 25
log 2 5
log 5 49 .
12) If sin(x) + cos(x) = 2 , find the value of sin3 x + cos3 x . Express your answer as a rational number in lowest terms.
13 Find the area of the region bounded by the lines x
x 2 y 11 and x – y 2.
2y
2, – 4 x
y
1,
14 Find the area of the circle that contains the point Q 9, 8 and that is tangent
to the line x – 2 y 2 at the point P 6, 2 .
Q
P
15) If log x y2
3, determine the value of log y x2 . Express your answer as a rational number in lowest terms.
16) When a complex number z is expressed the form z
of z is defined by
z
a2
a
b2 . Suppose that z
b
where a and b are real numbers, the modulus (or absolute value)
z
3
9 . Determine the value of z 2 .
17) Twenty balls numbered 1 to 20 are placed in a jar. Larry reaches into the jar and randomly removes two of the balls.
What is the probability that the sum of the numbers on the two removed balls is a multiple of 3? Express your answer
as a rational number in lowest terms.
18) The three vertices of a triangle are points on the graph of the parabola y
of the cubic equation x3
60 x2
153 x
1026
x2 . If the x-coordinates of the vertices are the roots
0, find the sum of the slopes of the three sides of the triangle.
19) For how many real numbers x will the mean of the set 6, 3, 10, 9, x be equal to the median?
20) Each side of square ABCD has length 3. Let M and N be points on sides BC and CD respectively such that BM
MAN. Find sin .
and let
ND
21) Find the value of the real number x such that 5 + x, 11 + x and 20 + x form a geometric progression in the given order.
22 Find the number of paths from the lower left corner to the
upper right corner of the given grid, if the only allowable moves
are along grid lines upward or to the right.
One such path is shown.
1
23) If x and z are real number such that 2
x –3
– 9 and
z
x
23 , find x + z.
z
1
24) If sin( ) = 4 , find sin(3 ). Express your answer as a rational number in lowest terms.
x– y
. Define the sequence an by a1
x y
25) Let f (x,y)
x2 – 4 – 4
26) Find the sum of all of the positive real solutions of
27 Triangle ABC is a 3
4
5 right triangle with AB
Construct the perpendicular AD1 and let AD1
perpendicular D1 D2 and let D1 D2
D2 D3 and let D2 D3
find
f (3,1) and an
f an , 1 for n
1
1.
4.
C
x1 . Construct the
x2 . Construct the perpendicular
D1
x3 . If this process is continued forever,
D3
xk .
k
x1
1
A
28) Let f (x)
3 x2 – x. Find all values of x such that f (f(x))
29 In triangle ABC, CAB
1. Find a2011 .
30 °, AC
2 and AB
5
x2
x3
B
D2
x.
B
3
Find BC.
5 3
A
30°
2
C
30) How many liters of a brine solution with a concentration of 30% salt should be added to 300 liters of brine with a concentration
of 23% salt so that the resulting solution has a concentration of 26% salt?
31) Four horses compete in a race. In how many different orders can the horses cross the finish line, assuming that all four horses
finish the race and that ties are possible?
32 As shown in the sketch, on each side of a square with side length 4, an
interior semicircle is drawn using that side as a diameter. Find the area of
the shaded region.
33) Two large pumps and one small pump can fill a swimming pool in 4 hours. One large pump and three small pumps can
fill the same swimming pool in 4 hours. How many minutes will it take four large pumps and four small pumps, working
together, to fill the swimming pool? (Assume that all large pumps pump at the rate R and all small pumps pump at the rate r.)
34) There are 40 students in the Travel Club. They discovered that 17 members have visited Mexico, 28 have
visited Canada, 10 have been to England, 12 have visited both Mexico and Canada, 3 have been only to
England and 4 have been only to Mexico. Some club members have not been to any of the three foreign
countries and an equal number have been to all three countries. How many students have been
to all three countries?
35) A bag contains 11 candy bars: three cost 50 cents each, four cost 1 dollar each and four cost 2 dollars each.
Three candy bars are randomally chosen from the bag, without replacement. What is the probability that the total
cost of the three candy bars is 4 dollars or more? Express your answer as a rational number in lowest terms.
36) If x4 + x3 + x2 + x + 1
0 and x +
1
1
> 0, determine the value of x + .
x
x
37) Suppose that a and r are real numbers such that the geometric series whose first term is a and whose ratio is r has a sum of 1
and the geometric series whose first term is a3 and whose ratio is r3 has a sum of 3. Find a.
38
y
Points A 0, 0 , B 18, 24 and C 11, 0 are the vertices of
triangle ABC. Point P is chosen in the interior of this triangle so that the
area of triangles ABP, APC and PBC are all equal. Find the coordinates of P.
Express your answer as an ordered pair x , y .
B 18,24
P
x
A 0,0
C 11,0
39 Find the distance between the centers of the inscribed and circumscribed circles
of a right triangle with sides of length 3, 4 and 5.
C
A
B
40) Let S be the set of all 11–digit binary sequences consisting of exactly two ones and nine zeros. For example,
00100000100 and 10000100000 are two of the elements of S. If each element of S is converted to a decimal
integer and all of these decimal integers are summed, what is the value of the sum? Express your answer
as an integer in base 10.
C
41 As shown in the sketch, circular arcs A C and B C have respective centers at
B and A. Suppose that S is a circle that is tangent to each of these arcs and also
to the line segment joining A and B. Find the radius of S if AB 24.
A
B
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-FOURTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 9, 2011
1) Evaluate 1
1
10
1
11
1
1
1
12
1
2011 2012
2011 2010
2011
2) Simplify the expression
1
13
. Express your answer as a rational number in lowest terms.
. Express your answer as a rational number in lowest terms.
3) The difference of two positive numbers is 4 and the product of the two numbers is 19. Find the sum of the
two numbers.
4) Find the value of a + b
2
if ab
– 14 and a2
b2
30 .
5 Consider the circle with diameter AD. If ABC 130 °, CDA
and BCA 20 °, find BAD. Express your answer in degrees.
50 °
C
B
A
D
6) If the length of a rectangle is increased by 40% and the width is decreased by 15%, what is the percentage change in the area of
the rectangle?
7) One day last month, Ray's Reasonably Reliable Repair Service offered the following Saturday special:
"Buy 3 shock absorbers at the regular price and receive an 80% discount on the fourth."
Jerome bought 4 shock absorbers on that day and paid a total of $176. What was the regular price of one shock absorber?
8) The two roots of the quadratic equation x2
85 x
9) For real numbers x, y and z, define F x, y, z
xy
c
0 are prime numbers. What is the value of c?
yz
z x. For which real numbers a is F 2, a, a
1
F 5, a, a
1?
10) Michelle has a collection of marbles, all of which are either blue or green. She is creating pairs of 1 blue marble and 1 green marble.
After a while, she notices that
2
3
of all the blue marbles are paired with
3
5
of all the green marbles. What fraction of Michelle’s
marble collection has been paired up? Express your answer as a rational number in lowest terms.
11) Find the integer value of the expression log 7
1
1
8
log 8 25
log 2 5
log 5 49 .
12) If sin(x) + cos(x) = 2 , find the value of sin3 x + cos3 x . Express your answer as a rational number in lowest terms.
13 Find the area of the region bounded by the lines x
x 2 y 11 and x – y 2.
2y
2, – 4 x
y
1,
14 Find the area of the circle that contains the point Q 9, 8 and that is tangent
to the line x – 2 y 2 at the point P 6, 2 .
Q
P
15) If log x y2
3, determine the value of log y x2 . Express your answer as a rational number in lowest terms.
16) When a complex number z is expressed the form z
of z is defined by
z
a2
a
b2 . Suppose that z
b
where a and b are real numbers, the modulus (or absolute value)
z
3
9 . Determine the value of z 2 .
17) Twenty balls numbered 1 to 20 are placed in a jar. Larry reaches into the jar and randomly removes two of the balls.
What is the probability that the sum of the numbers on the two removed balls is a multiple of 3? Express your answer
as a rational number in lowest terms.
18) The three vertices of a triangle are points on the graph of the parabola y
of the cubic equation x3
60 x2
153 x
1026
x2 . If the x-coordinates of the vertices are the roots
0, find the sum of the slopes of the three sides of the triangle.
19) For how many real numbers x will the mean of the set 6, 3, 10, 9, x be equal to the median?
20) Each side of square ABCD has length 3. Let M and N be points on sides BC and CD respectively such that BM
MAN. Find sin .
and let
ND
21) Find the value of the real number x such that 5 + x, 11 + x and 20 + x form a geometric progression in the given order.
22 Find the number of paths from the lower left corner to the
upper right corner of the given grid, if the only allowable moves
are along grid lines upward or to the right.
One such path is shown.
1
23) If x and z are real number such that 2
x –3
– 9 and
z
x
23 , find x + z.
z
1
24) If sin( ) = 4 , find sin(3 ). Express your answer as a rational number in lowest terms.
x– y
. Define the sequence an by a1
x y
25) Let f (x,y)
x2 – 4 – 4
26) Find the sum of all of the positive real solutions of
27 Triangle ABC is a 3
4
5 right triangle with AB
Construct the perpendicular AD1 and let AD1
perpendicular D1 D2 and let D1 D2
D2 D3 and let D2 D3
find
f (3,1) and an
f an , 1 for n
1
1.
4.
C
x1 . Construct the
x2 . Construct the perpendicular
D1
x3 . If this process is continued forever,
D3
xk .
k
x1
1
A
28) Let f (x)
3 x2 – x. Find all values of x such that f (f(x))
29 In triangle ABC, CAB
1. Find a2011 .
30 °, AC
2 and AB
5
x2
x3
B
D2
x.
B
3
Find BC.
5 3
A
30°
2
C
30) How many liters of a brine solution with a concentration of 30% salt should be added to 300 liters of brine with a concentration
of 23% salt so that the resulting solution has a concentration of 26% salt?
31) Four horses compete in a race. In how many different orders can the horses cross the finish line, assuming that all four horses
finish the race and that ties are possible?
32 As shown in the sketch, on each side of a square with side length 4, an
interior semicircle is drawn using that side as a diameter. Find the area of
the shaded region.
33) Two large pumps and one small pump can fill a swimming pool in 4 hours. One large pump and three small pumps can
fill the same swimming pool in 4 hours. How many minutes will it take four large pumps and four small pumps, working
together, to fill the swimming pool? (Assume that all large pumps pump at the rate R and all small pumps pump at the rate r.)
34) There are 40 students in the Travel Club. They discovered that 17 members have visited Mexico, 28 have
visited Canada, 10 have been to England, 12 have visited both Mexico and Canada, 3 have been only to
England and 4 have been only to Mexico. Some club members have not been to any of the three foreign
countries and an equal number have been to all three countries. How many students have been
to all three countries?
35) A bag contains 11 candy bars: three cost 50 cents each, four cost 1 dollar each and four cost 2 dollars each.
Three candy bars are randomally chosen from the bag, without replacement. What is the probability that the total
cost of the three candy bars is 4 dollars or more? Express your answer as a rational number in lowest terms.
36) If x4 + x3 + x2 + x + 1
0 and x +
1
1
> 0, determine the value of x + .
x
x
37) Suppose that a and r are real numbers such that the geometric series whose first term is a and whose ratio is r has a sum of 1
and the geometric series whose first term is a3 and whose ratio is r3 has a sum of 3. Find a.
38
y
Points A 0, 0 , B 18, 24 and C 11, 0 are the vertices of
triangle ABC. Point P is chosen in the interior of this triangle so that the
area of triangles ABP, APC and PBC are all equal. Find the coordinates of P.
Express your answer as an ordered pair x , y .
B 18,24
P
x
A 0,0
C 11,0
39 Find the distance between the centers of the inscribed and circumscribed circles
of a right triangle with sides of length 3, 4 and 5.
C
A
B
40) Let S be the set of all 11–digit binary sequences consisting of exactly two ones and nine zeros. For example,
00100000100 and 10000100000 are two of the elements of S. If each element of S is converted to a decimal
integer and all of these decimal integers are summed, what is the value of the sum? Express your answer
as an integer in base 10.
C
41 As shown in the sketch, circular arcs A C and B C have respective centers at
B and A. Suppose that S is a circle that is tangent to each of these arcs and also
to the line segment joining A and B. Find the radius of S if AB 24.
A
B