131 Contest 2014 Test
THE UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 19, 2014
1+
1) Express
1+
1
20
1
14
as a rational number in lowest terms.
2) Express 8 4ê3 I2 -3 - 9 -1ê2 M as a rational number in lowest terms.
3) Express log 2 H125L log 3 H49L log 5 H81L log 7 H64L as an integer.
4) The formula for limeade calls for 4 ounces of lime juice for every 12 ounces of water. Karla initially uses 18 ounces of water
to make her limeade. If her limeade has 40% too much lime juice, how many ounces of water does she need to add to her
mixture to have the correct ratio of lime juice to water? Express the answer as a rational number in lowest terms.
5) During the winter, 60% of Vermonters ski and during the summer, 45% of Vermonters hike. If 15% of Vermonters do both
activities, what percent of Vermonters do neither?
6L Find the total area of the shaded regions if the area of rectangle
ABCD is 40 square units.
A
B
D
C
7) A candy store sold bags of 40 caramels for $3.20, bags of 40 chocolates for $4.00 and mixed bags of chocolates
and caramels for $3.50. If the mixed bags also have 40 pieces of candy, how many caramels are in each mixed bag?
8) The function f satisfies 2 f HxL - 6 f J x M = x2 for all x ∫ 0. Find f (2) and express the answer as a rational number in lowest terms.
1
9) Find the real number x such that log 3 I x M + 4 log x H9L = 4.
10) Suppose that f is a function such that f H3 xL =
11) Express
12510 + 2510
1254 + 2511
in simplest form.
3
for all real x > 0. Determine the value of f H10L.
3+x
12) In how many ways can 24 cents be paid using any combination of pennies, nickels and dimes?
x - 2 -3 = 4.
13) Find all real values of x such that
14) The average of a set of 50 numbers is 45 and the average of a set of m numbers is 65. If the average of
the combined sets is 60, what is m?
15) Express sin2 H19 °L cos2 H26 °L +
1
2
sinH38 °L sinH52 °L + sin2 H26 °L cos2 H19 °L as a rational number in lowest terms.
1
16) The probability that Sheila hits the bullseye when playing darts is 4 . If she tosses three darts, what is the probability
she will hit the bullseye at least once? Express your answer as a rational number in lowest terms.
17) Find the coordinates of the center of the circle that passes through the points H7, 0L, H2, -1L and H2, -5L. Express
the answer as an ordered pair of real numbers Ha, bL.
18) If x and y satisfy
1
x
+
1
y
=
1
2
and x y = -6, find the value of x3 + y3 .
19) Ticket prices for a local community orchestra are $15 for adults, $12 for seniors and $7 for students. At a recent
concert, the orchestra sold 120 tickets for a total of $1481. What is the maximum possible number of student tickets
that were sold?
2014
20) Let a0 = 2, a1 = 5 and an = an-1 - an-2
for n ¥ 2.
Find
⁄ an .
n=0
21) What is the minimum value of
9 + 25 t6
, where t is a positive real number ?
t3
22) Let R be the region in the x y plane bounded by the line segments joining H0, 0L, H0, 5L, H4, 5L,
H4, 1L, H7, 1L, H7, 0L and H0, 0L, in the given order. The line y = k x divides R into two subregions of equal
area. Determine the value of k.
23) When a complex number z is expressed in the form z = a + b i, where i2 = -1 and a and b are real numbers,
the modulus of z, denoted z , is defined by z =
a2 + b2 . Find all complex numbers z of modulus 1
such that H3 + 4 iL z is a real number. Express your answer(s) in the form c + d i, where c and d are real numbers.
24L In D ABC, AB = AC and point Q Hstrictly between A and BL
A
is located on AB so that AQ = QC = CB. Determine the degree
measure of angle A.
Q
B
C
25) For a real number x, define f HxL =
16 x - x2 -
30 x - x2 - 224 . Determine the largest possible
positive value of f HxL.
26) In a list of the base 4 representations of the decimal integers from 0 to 1023, the digit 3 appears a total of k times. Find k.
C
27L Circles CP and CQ with centers at P and Q are externally
tangent and have radii 2 and 1, respectively. Line segment
T
AC is tangent to circle CP at A and line segment BC is tangent
S
to CP and CQ at T and S, respectively. Find the length AC.
A
B
P
Q
B
28L Suppose that A and B are points on a circle with
center O . If the perimeter of sector OAB is 10 units
and the area of sector OAB is 4 square units, find all
í
possible values of the length of arc AB .
O
A
29) If the roots of x2 + ax + b = 0 are the cubes of the roots of x2 + x + 2 = 0, find a and b.
30) How many positive integers x have the property that 14 is the remainder when 2014 is divided by x ?
31) Find the smallest positive value of x (in radians) such that tanH2 xL =
32L How many paths are there from A to B,
if at each intersection you can only move
in the indicated direction HsL ?
cosHxL - sinHxL
.
cosHxL + sinHxL
B
A
33) If x2 + 6 x - 6 = 0 , what is the value of x3 + 7 x2 + 2014 ?
34) For how many integers n is
20 - n
an integer ?
14 - n
35) The integer M consists of 500 threes and the integer N consists of 500 sixes. What is the sum of the digits
in the base 10 representation of the product M ÿ N ?
36) What is the smallest integer that is greater than I 5 +
6
3M ?
37) The geometric mean of a set of k positive real numbers 8x1 , x2 , x3 , ÿ ÿ ÿ , xk < is Hx1 ÿ x2 ÿ ÿ ÿ xk L 1êk .
Find the positive integer n such that the geometric mean of the set of all positive integer divisors of n is 70.
38L An equilateral triangle with side length one is
divided into four congruent triangles and the
central triangle is shaded. Let the shaded area be
A1 . The remaining three triangles are similarly
divided and each central triangle is shaded; the area
of the three shaded triangles is A2 . This process is
continued . The shaded areas A1 , A2 and A3 are
¶
shown. Find ⁄ An
n=1
39) How many different rectangles can be formed using edges in the left-hand figure below?
Two such rectangles are shown in the right-hand figure.
40L A grassy park in the shape of an equilateral
triangle is to be surrounded by a gravel
walkway whose outside edges form an equilateral
triangle. If the parallel sides of the walkway
are 2 meters apart and the area of the grassy park
is 30,000
2
2
3 square meters, what is the area
of the gravel walkway?
2
41L In a square of side length
2 , each side is divided into
10 equal pieces by inserting 9 equally spaced points. The
corresponding points on adjacent sides are connected by
straight line segments as indicated in the figure. Find the
sum of the lengths of these diagonal line segments.
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 19, 2014
1+
1) Express
1+
1
20
1
14
as a rational number in lowest terms.
2) Express 8 4ê3 I2 -3 - 9 -1ê2 M as a rational number in lowest terms.
3) Express log 2 H125L log 3 H49L log 5 H81L log 7 H64L as an integer.
4) The formula for limeade calls for 4 ounces of lime juice for every 12 ounces of water. Karla initially uses 18 ounces of water
to make her limeade. If her limeade has 40% too much lime juice, how many ounces of water does she need to add to her
mixture to have the correct ratio of lime juice to water? Express the answer as a rational number in lowest terms.
5) During the winter, 60% of Vermonters ski and during the summer, 45% of Vermonters hike. If 15% of Vermonters do both
activities, what percent of Vermonters do neither?
6L Find the total area of the shaded regions if the area of rectangle
ABCD is 40 square units.
A
B
D
C
7) A candy store sold bags of 40 caramels for $3.20, bags of 40 chocolates for $4.00 and mixed bags of chocolates
and caramels for $3.50. If the mixed bags also have 40 pieces of candy, how many caramels are in each mixed bag?
8) The function f satisfies 2 f HxL - 6 f J x M = x2 for all x ∫ 0. Find f (2) and express the answer as a rational number in lowest terms.
1
9) Find the real number x such that log 3 I x M + 4 log x H9L = 4.
10) Suppose that f is a function such that f H3 xL =
11) Express
12510 + 2510
1254 + 2511
in simplest form.
3
for all real x > 0. Determine the value of f H10L.
3+x
12) In how many ways can 24 cents be paid using any combination of pennies, nickels and dimes?
x - 2 -3 = 4.
13) Find all real values of x such that
14) The average of a set of 50 numbers is 45 and the average of a set of m numbers is 65. If the average of
the combined sets is 60, what is m?
15) Express sin2 H19 °L cos2 H26 °L +
1
2
sinH38 °L sinH52 °L + sin2 H26 °L cos2 H19 °L as a rational number in lowest terms.
1
16) The probability that Sheila hits the bullseye when playing darts is 4 . If she tosses three darts, what is the probability
she will hit the bullseye at least once? Express your answer as a rational number in lowest terms.
17) Find the coordinates of the center of the circle that passes through the points H7, 0L, H2, -1L and H2, -5L. Express
the answer as an ordered pair of real numbers Ha, bL.
18) If x and y satisfy
1
x
+
1
y
=
1
2
and x y = -6, find the value of x3 + y3 .
19) Ticket prices for a local community orchestra are $15 for adults, $12 for seniors and $7 for students. At a recent
concert, the orchestra sold 120 tickets for a total of $1481. What is the maximum possible number of student tickets
that were sold?
2014
20) Let a0 = 2, a1 = 5 and an = an-1 - an-2
for n ¥ 2.
Find
⁄ an .
n=0
21) What is the minimum value of
9 + 25 t6
, where t is a positive real number ?
t3
22) Let R be the region in the x y plane bounded by the line segments joining H0, 0L, H0, 5L, H4, 5L,
H4, 1L, H7, 1L, H7, 0L and H0, 0L, in the given order. The line y = k x divides R into two subregions of equal
area. Determine the value of k.
23) When a complex number z is expressed in the form z = a + b i, where i2 = -1 and a and b are real numbers,
the modulus of z, denoted z , is defined by z =
a2 + b2 . Find all complex numbers z of modulus 1
such that H3 + 4 iL z is a real number. Express your answer(s) in the form c + d i, where c and d are real numbers.
24L In D ABC, AB = AC and point Q Hstrictly between A and BL
A
is located on AB so that AQ = QC = CB. Determine the degree
measure of angle A.
Q
B
C
25) For a real number x, define f HxL =
16 x - x2 -
30 x - x2 - 224 . Determine the largest possible
positive value of f HxL.
26) In a list of the base 4 representations of the decimal integers from 0 to 1023, the digit 3 appears a total of k times. Find k.
C
27L Circles CP and CQ with centers at P and Q are externally
tangent and have radii 2 and 1, respectively. Line segment
T
AC is tangent to circle CP at A and line segment BC is tangent
S
to CP and CQ at T and S, respectively. Find the length AC.
A
B
P
Q
B
28L Suppose that A and B are points on a circle with
center O . If the perimeter of sector OAB is 10 units
and the area of sector OAB is 4 square units, find all
í
possible values of the length of arc AB .
O
A
29) If the roots of x2 + ax + b = 0 are the cubes of the roots of x2 + x + 2 = 0, find a and b.
30) How many positive integers x have the property that 14 is the remainder when 2014 is divided by x ?
31) Find the smallest positive value of x (in radians) such that tanH2 xL =
32L How many paths are there from A to B,
if at each intersection you can only move
in the indicated direction HsL ?
cosHxL - sinHxL
.
cosHxL + sinHxL
B
A
33) If x2 + 6 x - 6 = 0 , what is the value of x3 + 7 x2 + 2014 ?
34) For how many integers n is
20 - n
an integer ?
14 - n
35) The integer M consists of 500 threes and the integer N consists of 500 sixes. What is the sum of the digits
in the base 10 representation of the product M ÿ N ?
36) What is the smallest integer that is greater than I 5 +
6
3M ?
37) The geometric mean of a set of k positive real numbers 8x1 , x2 , x3 , ÿ ÿ ÿ , xk < is Hx1 ÿ x2 ÿ ÿ ÿ xk L 1êk .
Find the positive integer n such that the geometric mean of the set of all positive integer divisors of n is 70.
38L An equilateral triangle with side length one is
divided into four congruent triangles and the
central triangle is shaded. Let the shaded area be
A1 . The remaining three triangles are similarly
divided and each central triangle is shaded; the area
of the three shaded triangles is A2 . This process is
continued . The shaded areas A1 , A2 and A3 are
¶
shown. Find ⁄ An
n=1
39) How many different rectangles can be formed using edges in the left-hand figure below?
Two such rectangles are shown in the right-hand figure.
40L A grassy park in the shape of an equilateral
triangle is to be surrounded by a gravel
walkway whose outside edges form an equilateral
triangle. If the parallel sides of the walkway
are 2 meters apart and the area of the grassy park
is 30,000
2
2
3 square meters, what is the area
of the gravel walkway?
2
41L In a square of side length
2 , each side is divided into
10 equal pieces by inserting 9 equally spaced points. The
corresponding points on adjacent sides are connected by
straight line segments as indicated in the figure. Find the
sum of the lengths of these diagonal line segments.