128 Contest 2013 Test
THE UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SIXTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 13, 2013
1) Express
1
2
+
2
3
+
3
4
+
4
5
as a rational number in lowest terms.
8
2) Simplify the expression I 27 M
3) Express
4ê3 64 – 3ê2
I 81 M
.
Express your answer as a rational number in lowest terms.
log2 H7L ÿ log4 H9L ÿ log6 H5L
in simplest form.
log4 H5L ÿ log6 H7L ÿ log8 H3L
4) Larry and Jack are standing next to each other on a walking path. Jack begins walking north at a speed of 4 feet per second,
while Larry begins walking south at a speed of 3 feet per second. After 2 minutes, Jack turns around and begins walking south
at a speed of 5 feet per second. How far from the starting point will Larry have walked by the time Jack catches up to him?
5L Find the area of the polygon with vertices H1, 1L, H3, 2L, H2, 3L, H4, 4L
and H5, 0L. See the sketch. Express your answer as a rational number
in lowest terms.
4
3
2
1
1
2
3
4
5
6) To pass Mincle, a bridge troll, you must know his age and the number of his toes. He tells you that the product of these numbers
is 4378. You know that he is at least 150 years old and that bridge trolls do not live past 300. How many toes does he have?
7) Express 2013 as an integer in base 7.
8) Your two robots, Gort and Klaatu, can vacuum your house in 12 minutes if they work together. Gort can vacuum the house in 28
minutes if it works by itself. How long does it take Klaatu to vacuum the house by itself?
9) In Ms. Direction’s special topics class, 20% of the students are juniors and 80% are seniors. On a recent test,
the average score for the entire class was 85 and the average score for the seniors was 88.
What was the average score for the juniors ?
10) If
1
x
+
1
y
= 5 and
1
x
–
1
y
= 1, find the value x + y.
11) Find the value of 8 sinH15 °L cos3 H15 °L - 8 cosH15 °L sin3 H15 °L.
12) Given that
1
x
+
1
y
= 2 and xy = 6, find the value of x2 + y2 .
13) Larry has a sack containing 6 distinct objects. He draws one object and replaces it. He draws again, replaces the object and
draws a third time. What is the probability that he draws the same object exactly two times?
14L The area of a circle with center C is 72 p square units.
If the square ABCD has one vertex at C and the opposite
vertex A on the circle, what is the area of square ABCD?
15L An equilateral triangle with vertices A, B and C is inscribed in a
circle. If the perimeter of the triangle is 18 cm, find the length of
the arc of the circle between two adjacent vertices of the triangle.
C
B
D
A
C
A
B
P
16L Chord AB in circle C has length 12 cm. If M is the midpoint of
chord AB and P is a point on the circle such that PM is
3
A
perpendicular to AB and PM = 3 cm, find the diameter of
the circle.
M
6
6
B
17) Find the smallest real number c such that the equation | x – 2 | + |3x + 4 | = c has at least one solution.
8
18) If a, b, c and d are positive real numbers such that loga HbL = 9 , logb (c) = –
19) The sequence 8an < is defined by a0 = 2, a1 = 4 and an =
3
4
and logc HdL = 2 , find the value of logd HabcL.
6 aHn-1L
for n ¥ 2. Determine the value of a2013 .
aHn-2L
20) A bag contains 5 red marbles and 3 yellow marbles. Marbles are removed from the bag one at a time without
replacement until either all of the red marbles have been removed or all of the yellow marbles have been
removed. What is the probability that the last marble drawn from the bag is yellow ? Express your answer
as a rational number in lowest terms.
21) Suppose that x and y are positive real numbers such that logIx y3 M = 2 and log
x2
= 3.
y
Determine the value of logHx yL.
22) Find the number of positive integers k with 10 000 k 99 999 such that the middle digit is the average
of the first and fifth digits.
23) The sum of 54 consecutive positive integers is a perfect cube. What is the smallest possible value of the sum ?
24) Find the area of the region bounded by the graphs of
y + x + x = 10
.
y + x – x = –8
25) Find the value of a such that the three solutions of x3 – 8 x2 + ax – 12 = 0 are positive integers.
26) If
H 1 + i L2013
is expressed in the form a + b i where a and b are real numbers and i2 = – 1, find the value of a + b.
H 1 – i L2007
27) Six black checkers are placed on squares of a 6 by 6 checkerboard in the positions shown in Figure 1 and are left in place. A white
checker begins on the square at the lower left corner of the board (marked A in Figure 1) and follows a path from square to
square across the board, ending in the upper right corner of the board (marked B). How many different paths are there from
A to B if at each step the white checker can move one square to the right, one square up or one square diagonally upward to
the right and may not pass though any square occupied by a black checker? One such path is shown in Figure 2.
B
A
Figure 1
Figure 2
28L Let ABCD be a square of edge length 6. Using A B as
a diameter, draw a semicircle internal to the square. Using
D
C
D as the center and D A as a radius, draw a quartercircle
internal to the square. The semicircle and the quartercircle
intersect at E. What is the distance from E to A B ?
E
A
B
29L Let ABCD be a square of side length 16. A circle of radius r is
drawn through points C and D and is tangent to side AB. Find r.
D
C
A
B
2013
30) If
⁄ Ii k + i – k M
is expressed in the form a + b i where a and b are real numbers and i2 = – 1, find the value of a + b.
k= 0
31) Find all ordered pairs (x,y) that satisfy the system of equations
x2 + 4 x y - 8 x + 4 y2 - 16 y + 16 = 0
x y2 - 3 x y + 2 x - 2 y2 + 6 y - 4 = 0
.
32) If f (1) = 5 and f (n) = f (n –1) + 2n – 1 for all n ¥ 2, find f (100).
33) In a random arrangement of the letters of FREEZEDRIED , what is the probability that all of the vowels will be together in a single
consecutive grouping? Express your answer as a rational number in lowest terms.
34) Suppose that x and y are real numbers such that
2
2
value of x + y - 8 y ?
x + y + x - y = 10. What is the greatest possible
35L Let ABC be a nondegenerate right triangle. If AB = 30,
AD = 21, angle CAD = a and angle CDB = 2 a,
find CB.
C
α
2α
A
36L Let ABC be the triangle with vertices H0, 0L, H4, 0L
and H2, 3L. Find the coordinates of the point P that
is equidistant from A, B and C. Express your answer
as an ordered pair Hx, yL.
D
B
CH2,3L
PHx,yL
AH0,0L
BH4,0L
37) Suppose that a, b, c and d are positive integers such that
a b + 2 a + 2 b = 217
b c + 2 b + 2 c = 81
c d + 2 c + 2 d = 51
d a + 2 d + 2 a = 139.
Determine the value of a + b + c + d.
38) Let T be the triangular region whose vertices are H0, 0L, H4, 0L and H0, 5L . What is the probability that a randomly
chosen point from T is closer to H4, 2L than to H0, 0L ? Express your answer as a rational number in lowest terms.
39L Let S1 be a square of side length 3. Triangle T1 is formed by joining the
midpoint of the upper side of S1 to the endpoints of the lower side of S1 .
Let A1 be the area inside square S1 and outside triangle T1 . Square S2 is
inscribed in T1 with one side on the lower side of S1 and triangle T2
is formed by joining the midpoint of the upper side of S2 to the
endpoints of the lower side of S2 . Let A2 be the area inside square S2 and
outside triangle T2 . This process is successively repeated. The first four
¶
iterations are shown in the figure. Compute
⁄ Ak .
k= 1
40) Let t be the tens digit and u the units digit of 33
33
. Find t and u.
41L Three poles with circular cross sections are to be bound
together with a wire. The radii of the circular cross sections
are 1, 3 and 1 inches. The centers of the circles are on the
same straight line as indicated in the sketch. If the length of
the wire is written in the form a 3 + b p where a and b
are rational numbers, find a + b. Assume that the wire has
negligible thickness.
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SIXTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 13, 2013
1) Express
1
2
+
2
3
+
3
4
+
4
5
as a rational number in lowest terms.
8
2) Simplify the expression I 27 M
3) Express
4ê3 64 – 3ê2
I 81 M
.
Express your answer as a rational number in lowest terms.
log2 H7L ÿ log4 H9L ÿ log6 H5L
in simplest form.
log4 H5L ÿ log6 H7L ÿ log8 H3L
4) Larry and Jack are standing next to each other on a walking path. Jack begins walking north at a speed of 4 feet per second,
while Larry begins walking south at a speed of 3 feet per second. After 2 minutes, Jack turns around and begins walking south
at a speed of 5 feet per second. How far from the starting point will Larry have walked by the time Jack catches up to him?
5L Find the area of the polygon with vertices H1, 1L, H3, 2L, H2, 3L, H4, 4L
and H5, 0L. See the sketch. Express your answer as a rational number
in lowest terms.
4
3
2
1
1
2
3
4
5
6) To pass Mincle, a bridge troll, you must know his age and the number of his toes. He tells you that the product of these numbers
is 4378. You know that he is at least 150 years old and that bridge trolls do not live past 300. How many toes does he have?
7) Express 2013 as an integer in base 7.
8) Your two robots, Gort and Klaatu, can vacuum your house in 12 minutes if they work together. Gort can vacuum the house in 28
minutes if it works by itself. How long does it take Klaatu to vacuum the house by itself?
9) In Ms. Direction’s special topics class, 20% of the students are juniors and 80% are seniors. On a recent test,
the average score for the entire class was 85 and the average score for the seniors was 88.
What was the average score for the juniors ?
10) If
1
x
+
1
y
= 5 and
1
x
–
1
y
= 1, find the value x + y.
11) Find the value of 8 sinH15 °L cos3 H15 °L - 8 cosH15 °L sin3 H15 °L.
12) Given that
1
x
+
1
y
= 2 and xy = 6, find the value of x2 + y2 .
13) Larry has a sack containing 6 distinct objects. He draws one object and replaces it. He draws again, replaces the object and
draws a third time. What is the probability that he draws the same object exactly two times?
14L The area of a circle with center C is 72 p square units.
If the square ABCD has one vertex at C and the opposite
vertex A on the circle, what is the area of square ABCD?
15L An equilateral triangle with vertices A, B and C is inscribed in a
circle. If the perimeter of the triangle is 18 cm, find the length of
the arc of the circle between two adjacent vertices of the triangle.
C
B
D
A
C
A
B
P
16L Chord AB in circle C has length 12 cm. If M is the midpoint of
chord AB and P is a point on the circle such that PM is
3
A
perpendicular to AB and PM = 3 cm, find the diameter of
the circle.
M
6
6
B
17) Find the smallest real number c such that the equation | x – 2 | + |3x + 4 | = c has at least one solution.
8
18) If a, b, c and d are positive real numbers such that loga HbL = 9 , logb (c) = –
19) The sequence 8an < is defined by a0 = 2, a1 = 4 and an =
3
4
and logc HdL = 2 , find the value of logd HabcL.
6 aHn-1L
for n ¥ 2. Determine the value of a2013 .
aHn-2L
20) A bag contains 5 red marbles and 3 yellow marbles. Marbles are removed from the bag one at a time without
replacement until either all of the red marbles have been removed or all of the yellow marbles have been
removed. What is the probability that the last marble drawn from the bag is yellow ? Express your answer
as a rational number in lowest terms.
21) Suppose that x and y are positive real numbers such that logIx y3 M = 2 and log
x2
= 3.
y
Determine the value of logHx yL.
22) Find the number of positive integers k with 10 000 k 99 999 such that the middle digit is the average
of the first and fifth digits.
23) The sum of 54 consecutive positive integers is a perfect cube. What is the smallest possible value of the sum ?
24) Find the area of the region bounded by the graphs of
y + x + x = 10
.
y + x – x = –8
25) Find the value of a such that the three solutions of x3 – 8 x2 + ax – 12 = 0 are positive integers.
26) If
H 1 + i L2013
is expressed in the form a + b i where a and b are real numbers and i2 = – 1, find the value of a + b.
H 1 – i L2007
27) Six black checkers are placed on squares of a 6 by 6 checkerboard in the positions shown in Figure 1 and are left in place. A white
checker begins on the square at the lower left corner of the board (marked A in Figure 1) and follows a path from square to
square across the board, ending in the upper right corner of the board (marked B). How many different paths are there from
A to B if at each step the white checker can move one square to the right, one square up or one square diagonally upward to
the right and may not pass though any square occupied by a black checker? One such path is shown in Figure 2.
B
A
Figure 1
Figure 2
28L Let ABCD be a square of edge length 6. Using A B as
a diameter, draw a semicircle internal to the square. Using
D
C
D as the center and D A as a radius, draw a quartercircle
internal to the square. The semicircle and the quartercircle
intersect at E. What is the distance from E to A B ?
E
A
B
29L Let ABCD be a square of side length 16. A circle of radius r is
drawn through points C and D and is tangent to side AB. Find r.
D
C
A
B
2013
30) If
⁄ Ii k + i – k M
is expressed in the form a + b i where a and b are real numbers and i2 = – 1, find the value of a + b.
k= 0
31) Find all ordered pairs (x,y) that satisfy the system of equations
x2 + 4 x y - 8 x + 4 y2 - 16 y + 16 = 0
x y2 - 3 x y + 2 x - 2 y2 + 6 y - 4 = 0
.
32) If f (1) = 5 and f (n) = f (n –1) + 2n – 1 for all n ¥ 2, find f (100).
33) In a random arrangement of the letters of FREEZEDRIED , what is the probability that all of the vowels will be together in a single
consecutive grouping? Express your answer as a rational number in lowest terms.
34) Suppose that x and y are real numbers such that
2
2
value of x + y - 8 y ?
x + y + x - y = 10. What is the greatest possible
35L Let ABC be a nondegenerate right triangle. If AB = 30,
AD = 21, angle CAD = a and angle CDB = 2 a,
find CB.
C
α
2α
A
36L Let ABC be the triangle with vertices H0, 0L, H4, 0L
and H2, 3L. Find the coordinates of the point P that
is equidistant from A, B and C. Express your answer
as an ordered pair Hx, yL.
D
B
CH2,3L
PHx,yL
AH0,0L
BH4,0L
37) Suppose that a, b, c and d are positive integers such that
a b + 2 a + 2 b = 217
b c + 2 b + 2 c = 81
c d + 2 c + 2 d = 51
d a + 2 d + 2 a = 139.
Determine the value of a + b + c + d.
38) Let T be the triangular region whose vertices are H0, 0L, H4, 0L and H0, 5L . What is the probability that a randomly
chosen point from T is closer to H4, 2L than to H0, 0L ? Express your answer as a rational number in lowest terms.
39L Let S1 be a square of side length 3. Triangle T1 is formed by joining the
midpoint of the upper side of S1 to the endpoints of the lower side of S1 .
Let A1 be the area inside square S1 and outside triangle T1 . Square S2 is
inscribed in T1 with one side on the lower side of S1 and triangle T2
is formed by joining the midpoint of the upper side of S2 to the
endpoints of the lower side of S2 . Let A2 be the area inside square S2 and
outside triangle T2 . This process is successively repeated. The first four
¶
iterations are shown in the figure. Compute
⁄ Ak .
k= 1
40) Let t be the tens digit and u the units digit of 33
33
. Find t and u.
41L Three poles with circular cross sections are to be bound
together with a wire. The radii of the circular cross sections
are 1, 3 and 1 inches. The centers of the circles are on the
same straight line as indicated in the sketch. If the length of
the wire is written in the form a 3 + b p where a and b
are rational numbers, find a + b. Assume that the wire has
negligible thickness.