129 Contest 2014 Solutions
THE UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 19, 2014
1) Express
1
1
1
1
20
1
1
as a rational number in lowest terms.
14
1
140
140
20
1
14
140 7
140 10
2) Express 8 4 3 2
8 4 3 2
3
9
12
49
147
150
3
9
50
12
16 8
1
as a rational number in lowest terms.
1
3
16
3 8
24
16 5
24
20
6
10
3
3) Express log 2 125 log 3 49 log 5 81 log 7 64 as an integer.
log 2 125 log 3 49 log 5 81 log 7 64
3 log2 5 2 log3 7 4 log5 3 6 log7 2
log 2 125 log 3 49 log 5 81 log 7 64
3 log2 5 2
log 2 125 log 3 49 log 5 81 log 7 64
3 2 4 6 144
log2 7
log2 3
4
log2 3
log2 5
6
1
log2 7
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.
Original lime
18 x 3
42
5
x 3
18
42
5
18
12
4 1.4
126
5
90
5
3
2
4
14
5
42
5
36
5
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?
Ski
45
Hike
15
30
10
45 15 30 x 100
x
10
6 Find the total area of the shaded regions if the area of rectangle
ABCD is 40 square units.
A
12x
B
12y
y
D
2 2 2 x 2 y 4 x y 4 40
1
1
1
1
1
x
10
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?
x
3.20
40
40 x
4.00
40
350
320 x
40 x 400 350 40
320 x
40 400
400 x 40(350)
80 x 40 350 400
40 50
80
x
40 50
25
8) The function f satisfies 2 f x
6 fx
1
x2 for all x 0. Find f (2) and express the
answer as a rational number in lowest terms.
6 f2 4
1
2f 2
2 f2 6 f 2
1
1
4
3 f2 2
1
f 2
f2 3 f 2
1
3 f2 2
1
f 2
3 f2 9 f 2
1
3
8
19
8
8f 2
f 2
1
8
–
19
64
9) Find the real number x such that log 3
1
2
log3 x
8 logx 3
1
2
log3 x
1
8 log3 x
1
2
y
8
y
4
4
4
x
4 log x 9
4.
C
16
y
y
y2
8
8 y 16 0
y 4
2
0
y 4
log3 x
4
81
x
3
3 x
10) Suppose that f is a function such that f 3 x
for all real x 0.
Determine the value of f 10 .
f 3
f 10
10
3
2510
2511
9
9 10
10
3
12510
1254
11) Express
12510
1254
3
3
2510
2511
530
512
9
19
in simplest form.
520 510 1
520
522
58
512 510 1
54
625
12) In how many ways can 24 cents be paid using any combination of pennies,
nickels and dimes?
10
20
10
10
10
0
0
0
0
0
5
0
0
5
10
0
5
10
15
20
1
9
14
9
4
24
19
14
9
4
9
n
x 2
13) Find all real values of x such that
||x–2|–3|=4
3
|x–2|–3 =4
|x–2|=7
4.
or
|x–2|–3 =–4
| x – 2 | = – 1 Impossible
x – 2 = 7 or x – 2 = – 7
x = 9 or x – 5
13) If the number 15! written in base 12 ends in k zeros, what is the value of k?
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?
50 45
m 65
50 m
50 45
60
65 m 50 60
60 m
5 m 50 15
m 3 150
150
15) Express sin2 19
cos2 26
1
2
sin 38
sin 52
sin2 26
sin2 26
cos2 19
cos2 19
number in lowest terms.
sin2 19
sin2 19
cos2 26
cos2 26
1
2
sin 38
1
2
sin 52
sin 2 19
sin 2 26
sin2 26
cos2 19
as a rational
sin2 19
cos2 26
1
2
sin2 19
cos2 26
2 sin 19
2 sin 19
cos 19
cos 19
2 sin 26
sin 26
sin2 26
cos 26
sin2 26
cos 26
sin 19
cos 26
sin 45
2
1
sin 26
2
2
cos2 19
cos2 19
cos 19
2
1
2
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.
Hits 1 3 14 34
27
64
Hits 2 3 14 34
9
64
2
2
14
3
Hits 3
1
64
37
64
37
p
64
17) Find the coordinates of the center of the circle that passes through the points
7, 0 , 2, 1 and 2, 5 . Express the answer as an ordered pair of real numbers a, b .
7 h2
2 h2
2 h2
1
2
3
0 k 2 r2
1 k 2 r2
5 k 2 r2
49 14 h h2 k2 r 2
4 4 h h 2 1 2 k k2 r 2
4 4 h h2 25 10 k k2 r 2
2
1
3
2
45 10 h 1 2 k 0
24 8 k 0
h 5
5, 3
h, k
18) If x and y satisfy
1
x
1
y
x y 3
xy
1
2
1
x
y3
x3
y3
1
2
and xy
1
2
x y
1
2
6, find the value of x3
xy
1
2
6
3
1
8
3 x2 y
x3
1
y
x y
xy
x3 3 x2 y 3 xy 2 y 3
xy 3
x3
3
k
3 xy2
3 xy x y
63
8
1
8
y3
1
8
3 xy x y
1
8
6
6
2
3
666
222
3
6
3
27 54
81
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?
Let A, S and T be the numbers of adult, senior and student tickets sold.
A S T 120
15 A 12 S 7 T 1481
Solving for A and S
41 5 T
3
A
3 Α and 319 8 T
41 5 T
319 8 T
0
and S
319 8 T
3
3 Β for non negative integers Α and Β
39
T
T 41 5T 319–8T
39 236
7
38 231
15
Thus T
20) Let a0
38
2, a1
5 and an
an
1
an
for n 2.
2
Find
2014
n
a0
2
a1
5
a2
3
a3
2
a4
1
a5
1
a6
0
a7
1
a8
1
a9
0
an .
0
Thus, starting with a4 each 3 consecutive terms asum to 2.
2014
2014
an
n
a0
a1
a2
2013
a3
0
an
n
a0
a1
a2
a3
4
an
n
a2014
4
2014
an
n
2
5
3
2
670 2
1
1353
0
21) What is the minimum value of
9 25 t6
t3
9
2
5 t3
t3
30 t3
9
2
5 t3
t3
9 25 t6
,
t3
where t is a positive real number ?
30
Thus the minimum value occurs when 9
5 t3
0.
Minimum = 30
22) Let R be the region in the x y plane bounded by the line segments joining
0, 0 , 0, 5 , 4, 5 , 4, 1 , 7, 1 , 7, 0 and 0, 0 , in the given order.
The line y k x divides R into two subregions of equal area. Determine the value of k.
5
4
3
2
0, 5
4, 5
5
4
3
2
1
7, 1
4, 1
2
0, 0
4
Total area = 5 4
3 1
6
23
From the upper trapezoid 4
4 10 4 k
23
7, 0
5
5 4k
2
23
2
40 16 k 23
16 k 17
k
23) When a complex number z is expressed in the form z a
and a and b are real numbers, the modulus of z, denoted
a2
z
3 4 i a bi
3a 4b
25
16
b2
1
a bi
b i, where i2
1
z , is defined by
a2
b2
3
4
5
5
b2
i or
1
16
25
4
5
5
d i, where c and d are
3
4
3
4
a
2
b
b2
1
b
16
9
1 b2
1
4
5
b
3
i
A
24 In ABC, AB AC and point Q strictly between A and B
is located on AB so that AQ QC CB. Determine the degree
measure of angle A.
Α
Q
Β
B
AQ QC
QAC
QC CB
CQB
4i z
4a 3b i
For the product to be real 4 a 3 b 0
1
16
b2 . Find all complex numbers z of modulus 1 such that 3
is a real number. Express your answer(s) in the form c
real numbers.
z
17
QCA
Α
CBQ Β
Β
Α
Β Α
C
AB AC
ABC
Β an exterior angle of
ACB Β
QAC
QCB Β Β Β
From
Α 180
QCB Β
Β
Α
2Α
3 Β Α 180
3 2Α
5 Α 180
a 180
Α 36
25) For a real number x, define f x
16 x x2
30 x
Determine the largest possible positive value of f x .
x2
16 x
30 x
x2
224
x 16 x
x2
224 .
x 14 x 16
Both radicands are positive for 14 x 16. Max occurs when the first is
largest and the second smallest. i.e. x 14
f 14
14 16 14
14 2
2
7
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.
33 3334
The decimal integers 0 to 1023 in base 4 can be represented as 00 0004
Considering each of the base 4 integers as 5 digits, the total number of digits is 5(1024).
Since each digit appears an equal number of times, the number of 3s is
tangent and have radii 2 and 1, respectively. Line segment
AC is tangent to circle CP at A and line segment BC is tangent
to CP and CQ at T and S, respectively. Find the length AC.
T
S
2
A
2
P
Q S B sin Β
1
1 x
From
P T B sin Β
2
4 x
1
1 x
From
From
2
4 x
4 x 2 2x
x 2
Q S B sin Β
1
1 2
1
3
tan Β
AB C tan Β
AC
6 x
AC
8
1
8
1
8
AC
1280
C
27 Circles CP and CQ with centers at P and Q are externally
From
5 1024
4
8
8
2
2
8
Θ
1
1
Q
x
Β
B
B
28 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
possible values of the length of arc AB .
r
s
Θ
O
A
arc AB s r Θ
rΘ
10 2 r
r2 Θ
1
2
4
5
rΘ
2
10 r Θ
2
r
r rΘ
rΘ
8
rΘ
2
10 r Θ
8
29) If the roots of x2
r1 3 and r2 3 roots of x2
r1
r2
r1 3
r1 r2
3
3
r1 3
r2 3
2
rΘ 8 rΘ 2
16
0
23
x 2 0
r1 r2
ax b 0
r1 3 r2 3
3 r1 r2 3
r2 3
1
1
r1 3
r2 3
x 2 0, find a and b.
2 and r1
r1 3
b and
2, 8
rΘ
ax b 0 are the cubes of the roots of x2
Let r1 and r2 be the roots of x2
r1 3 r2 3
rΘ
2
5
1
r2
r2 3 a
8
3 r1 2 r2
3 r1 r2 r1
r2
1 3 2
1
5
5, 8
a, b
30) How many positive integers x have the property that 14 is the remainder when
2014 is divided by x ?
2014 qx 14
4
2000 2 5
qx 2000 so x divides 2000 and x 14.
3
4 1 3 1
20 divisors. Of these 1, 2, 4, 8, 5, 10 are less than 14.
The number of x satisfying the given conditions is 20 6 14
31) Find the smallest positive value of x (in radians) such that tan 2 x
tan 2 x
cos x
cos x
cos x
cos x
sin 2 x
cos 2 x
1 sin 2 x
cos 2 x
1
sin x
sin x
sin x
sin x
sin 2 x
cos 2 x
sin 2 x
cos x
cos x
sin x
sin x
sin 2 x
cos x
cos x
sin x
sin x
cos2 x
2 sin 2 x
1
2 sin x cos x sin2 x
cos2 x sin2 x
sin 2 x
1
2
2x
Π
6
x
Π
12
.
B
32 How many paths are there from A to B,
if at each intersection you can only move
in the indicated direction s ?
90
1
89
1
1
6
82
4
A1
16
2
60
6
22
22
Number of paths = 90
33) If x2
x
2
6 x 6 0 , what is the value of x3
6x 6 0
x
2
7 x2
2014 xx2
x3
7 x2
2014 x 6 x
3
2
7x
6 x x 2014
2014
2014 6 2014 2020
20 n
34) For how many integers n is
20 n
14 n
2014?
6x 6
x3
x
7 x2
an integer?
14 n
6
1 14 n
14 n divides 6
14 n
1, 2, 3, 6
Thus 8 values of 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 N ?
M N
10500
3
M N
2
9
1
2
10500
3
101000
M N 2
2
9
10500
1 2 10500
2
101000 1
9
M N 2 111
M N 222
1
2
1
1
10500 1
9
1 2 111
2 – 444
222
222
– 444
2222
4444
222
217
7778
4
1
1000 2s minus 500 4s
From left to right: 499 twos, 1 one, 499 sevens and 1 eight.
499 2
1 499 7
8 499 2 7
9 499 9
9 500 9
4500
6
36) What is the smallest integer that is greater than
6
6
5
4
5
2
3
3
?
3
2
4
5
6
5
3
6
5
6
6 5 3
15 5 3
2
20 5 3
3
15 5 3
4
6 5 3
5
3
6
5
3
6
5
6
6 5 3
15 5 3
2
20 5 3
3
15 5 3
4
6 5 3
5
3
6
5
3
5
6
5
5
3
6
2 5
6
5
3
5
3
4
4
15 5 3
6
4
0
3
2
2
15 5 3
2
2
0
4
0
3
6
6
5
3
5
3
2 125 12 25 3 15 5 9
6
27
6
2 125 1125 675 27
6
Since
3
5
3
2 1952
3904
6
1
5
3
3904
37) The geometric mean of a set of k positive real numbers x1 , x2 , x3 ,
, xk is
1k
x1 x2
xk . Find the positive integer n such that the geometric mean of the
set of all positive integer divisors of n is 70.
If n has prime factorization n p1 Α1 p2 Α2
k Α1 1
Α2 1
Αj 1 .
If all of the Αi are even, k is odd and
divisor
n . Then the product
k 1
2
x1 x2
pj Αj , the number of divisors of n is
of the divisors of n appear in pairs xi and
xk
nk
1 2
n1 2
n
k2
and x1 x2
xk
If not all of the Αi are even, k is even and the divisors of n appear in pairs xi and
Then the product
x1 x2
Thus n1 2
n 4900
70
xk
nk 2 and x1 x2
xk
1k
n
x1
1k
n
x1
n1 2 .
38 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
The area of an equilateral triangle of side length S is A
s2
4
3 . Assuming that the
original triangle has side lenght S,
The first shaded triangle has side length
S
2
and A1
The three shaded triangles in A2 have side length
S
4
2
S 2
3
4
and A2
4
S 2 3
3
4
with the remaining
n1 2 .
The nine shaded triangles in A3 have side length
S
8
and A3
8
S 2 32
3
4
Continuing this pattern:
"
! An
n 1
"
! An
n 1
"
! An
n 1
2
3
4
S2
42
3 1
S2
42
3
S 2
4
S 2 3
s2
8
S 2 32
3
4
Using the sum of the geormetric series with r
32
24
22
1
1
With s 1
3
3
4
3
4
3
4
3
4
3
An
4
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.
0
0
0
0
0
0
0
6
5
4
3
2
1
0
12
10
8
6
4
2
0
15
12
9
0
6
3
0
21
17
13
3
8
4
0
27
22
17
12
10
5
0
75
66
51
24
30
15
Labeling each node with the number of rectangles with the node as lower left
corner and adding:
75 66 51 24 30 15 261
40 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 3 square meters, what is the area
of the gravel walkway ?
2
2
2
2
2
S2
2
S1
2 3
4
30
2
Let the sides of the inside and outside triantles be S1 and S2 respectively.
The areas are respectively
Aw
3
4
S2
3
4
2
3
4
S1 2 and
3
4
S2 2 . Hence the area of the walkway is
S1 2 .
From the diagram above we can see that S2
S1
4
3.
2
S1 4
3
Aw
4
Aw
3 2
3
3 S1
S1
4
2
3
3
4
S1
2
3 2
3 200
3 S1
48
S1
2
12
Using the given area of the inside triangle:
Aw
8
3
12
3
4
S1
3 1200 12
2
30, 000
1212
3
3
S1
200
3
41 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.
s
10
s
L3
10
L2
s
L1
10
L1 =
10 2
10 2 =
s
10
2
L2 =
10
10
=
2s
10
2
Lk =
10
10
=
ks
10
2
s
2s 2
ks 2
9
T=2
k
s=
2
1
s
2s 2
ks 2
Lk + L10 = 2
T = 20
2
s
10
k = 1, 2, 3,
9 10
2
1 = 10
10
2 s
DEPARTMENT OF MATHEMATICS AND STATISTICS
FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 19, 2014
1) Express
1
1
1
1
20
1
1
as a rational number in lowest terms.
14
1
140
140
20
1
14
140 7
140 10
2) Express 8 4 3 2
8 4 3 2
3
9
12
49
147
150
3
9
50
12
16 8
1
as a rational number in lowest terms.
1
3
16
3 8
24
16 5
24
20
6
10
3
3) Express log 2 125 log 3 49 log 5 81 log 7 64 as an integer.
log 2 125 log 3 49 log 5 81 log 7 64
3 log2 5 2 log3 7 4 log5 3 6 log7 2
log 2 125 log 3 49 log 5 81 log 7 64
3 log2 5 2
log 2 125 log 3 49 log 5 81 log 7 64
3 2 4 6 144
log2 7
log2 3
4
log2 3
log2 5
6
1
log2 7
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.
Original lime
18 x 3
42
5
x 3
18
42
5
18
12
4 1.4
126
5
90
5
3
2
4
14
5
42
5
36
5
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?
Ski
45
Hike
15
30
10
45 15 30 x 100
x
10
6 Find the total area of the shaded regions if the area of rectangle
ABCD is 40 square units.
A
12x
B
12y
y
D
2 2 2 x 2 y 4 x y 4 40
1
1
1
1
1
x
10
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?
x
3.20
40
40 x
4.00
40
350
320 x
40 x 400 350 40
320 x
40 400
400 x 40(350)
80 x 40 350 400
40 50
80
x
40 50
25
8) The function f satisfies 2 f x
6 fx
1
x2 for all x 0. Find f (2) and express the
answer as a rational number in lowest terms.
6 f2 4
1
2f 2
2 f2 6 f 2
1
1
4
3 f2 2
1
f 2
f2 3 f 2
1
3 f2 2
1
f 2
3 f2 9 f 2
1
3
8
19
8
8f 2
f 2
1
8
–
19
64
9) Find the real number x such that log 3
1
2
log3 x
8 logx 3
1
2
log3 x
1
8 log3 x
1
2
y
8
y
4
4
4
x
4 log x 9
4.
C
16
y
y
y2
8
8 y 16 0
y 4
2
0
y 4
log3 x
4
81
x
3
3 x
10) Suppose that f is a function such that f 3 x
for all real x 0.
Determine the value of f 10 .
f 3
f 10
10
3
2510
2511
9
9 10
10
3
12510
1254
11) Express
12510
1254
3
3
2510
2511
530
512
9
19
in simplest form.
520 510 1
520
522
58
512 510 1
54
625
12) In how many ways can 24 cents be paid using any combination of pennies,
nickels and dimes?
10
20
10
10
10
0
0
0
0
0
5
0
0
5
10
0
5
10
15
20
1
9
14
9
4
24
19
14
9
4
9
n
x 2
13) Find all real values of x such that
||x–2|–3|=4
3
|x–2|–3 =4
|x–2|=7
4.
or
|x–2|–3 =–4
| x – 2 | = – 1 Impossible
x – 2 = 7 or x – 2 = – 7
x = 9 or x – 5
13) If the number 15! written in base 12 ends in k zeros, what is the value of k?
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?
50 45
m 65
50 m
50 45
60
65 m 50 60
60 m
5 m 50 15
m 3 150
150
15) Express sin2 19
cos2 26
1
2
sin 38
sin 52
sin2 26
sin2 26
cos2 19
cos2 19
number in lowest terms.
sin2 19
sin2 19
cos2 26
cos2 26
1
2
sin 38
1
2
sin 52
sin 2 19
sin 2 26
sin2 26
cos2 19
as a rational
sin2 19
cos2 26
1
2
sin2 19
cos2 26
2 sin 19
2 sin 19
cos 19
cos 19
2 sin 26
sin 26
sin2 26
cos 26
sin2 26
cos 26
sin 19
cos 26
sin 45
2
1
sin 26
2
2
cos2 19
cos2 19
cos 19
2
1
2
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.
Hits 1 3 14 34
27
64
Hits 2 3 14 34
9
64
2
2
14
3
Hits 3
1
64
37
64
37
p
64
17) Find the coordinates of the center of the circle that passes through the points
7, 0 , 2, 1 and 2, 5 . Express the answer as an ordered pair of real numbers a, b .
7 h2
2 h2
2 h2
1
2
3
0 k 2 r2
1 k 2 r2
5 k 2 r2
49 14 h h2 k2 r 2
4 4 h h 2 1 2 k k2 r 2
4 4 h h2 25 10 k k2 r 2
2
1
3
2
45 10 h 1 2 k 0
24 8 k 0
h 5
5, 3
h, k
18) If x and y satisfy
1
x
1
y
x y 3
xy
1
2
1
x
y3
x3
y3
1
2
and xy
1
2
x y
1
2
6, find the value of x3
xy
1
2
6
3
1
8
3 x2 y
x3
1
y
x y
xy
x3 3 x2 y 3 xy 2 y 3
xy 3
x3
3
k
3 xy2
3 xy x y
63
8
1
8
y3
1
8
3 xy x y
1
8
6
6
2
3
666
222
3
6
3
27 54
81
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?
Let A, S and T be the numbers of adult, senior and student tickets sold.
A S T 120
15 A 12 S 7 T 1481
Solving for A and S
41 5 T
3
A
3 Α and 319 8 T
41 5 T
319 8 T
0
and S
319 8 T
3
3 Β for non negative integers Α and Β
39
T
T 41 5T 319–8T
39 236
7
38 231
15
Thus T
20) Let a0
38
2, a1
5 and an
an
1
an
for n 2.
2
Find
2014
n
a0
2
a1
5
a2
3
a3
2
a4
1
a5
1
a6
0
a7
1
a8
1
a9
0
an .
0
Thus, starting with a4 each 3 consecutive terms asum to 2.
2014
2014
an
n
a0
a1
a2
2013
a3
0
an
n
a0
a1
a2
a3
4
an
n
a2014
4
2014
an
n
2
5
3
2
670 2
1
1353
0
21) What is the minimum value of
9 25 t6
t3
9
2
5 t3
t3
30 t3
9
2
5 t3
t3
9 25 t6
,
t3
where t is a positive real number ?
30
Thus the minimum value occurs when 9
5 t3
0.
Minimum = 30
22) Let R be the region in the x y plane bounded by the line segments joining
0, 0 , 0, 5 , 4, 5 , 4, 1 , 7, 1 , 7, 0 and 0, 0 , in the given order.
The line y k x divides R into two subregions of equal area. Determine the value of k.
5
4
3
2
0, 5
4, 5
5
4
3
2
1
7, 1
4, 1
2
0, 0
4
Total area = 5 4
3 1
6
23
From the upper trapezoid 4
4 10 4 k
23
7, 0
5
5 4k
2
23
2
40 16 k 23
16 k 17
k
23) When a complex number z is expressed in the form z a
and a and b are real numbers, the modulus of z, denoted
a2
z
3 4 i a bi
3a 4b
25
16
b2
1
a bi
b i, where i2
1
z , is defined by
a2
b2
3
4
5
5
b2
i or
1
16
25
4
5
5
d i, where c and d are
3
4
3
4
a
2
b
b2
1
b
16
9
1 b2
1
4
5
b
3
i
A
24 In ABC, AB AC and point Q strictly between A and B
is located on AB so that AQ QC CB. Determine the degree
measure of angle A.
Α
Q
Β
B
AQ QC
QAC
QC CB
CQB
4i z
4a 3b i
For the product to be real 4 a 3 b 0
1
16
b2 . Find all complex numbers z of modulus 1 such that 3
is a real number. Express your answer(s) in the form c
real numbers.
z
17
QCA
Α
CBQ Β
Β
Α
Β Α
C
AB AC
ABC
Β an exterior angle of
ACB Β
QAC
QCB Β Β Β
From
Α 180
QCB Β
Β
Α
2Α
3 Β Α 180
3 2Α
5 Α 180
a 180
Α 36
25) For a real number x, define f x
16 x x2
30 x
Determine the largest possible positive value of f x .
x2
16 x
30 x
x2
224
x 16 x
x2
224 .
x 14 x 16
Both radicands are positive for 14 x 16. Max occurs when the first is
largest and the second smallest. i.e. x 14
f 14
14 16 14
14 2
2
7
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.
33 3334
The decimal integers 0 to 1023 in base 4 can be represented as 00 0004
Considering each of the base 4 integers as 5 digits, the total number of digits is 5(1024).
Since each digit appears an equal number of times, the number of 3s is
tangent and have radii 2 and 1, respectively. Line segment
AC is tangent to circle CP at A and line segment BC is tangent
to CP and CQ at T and S, respectively. Find the length AC.
T
S
2
A
2
P
Q S B sin Β
1
1 x
From
P T B sin Β
2
4 x
1
1 x
From
From
2
4 x
4 x 2 2x
x 2
Q S B sin Β
1
1 2
1
3
tan Β
AB C tan Β
AC
6 x
AC
8
1
8
1
8
AC
1280
C
27 Circles CP and CQ with centers at P and Q are externally
From
5 1024
4
8
8
2
2
8
Θ
1
1
Q
x
Β
B
B
28 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
possible values of the length of arc AB .
r
s
Θ
O
A
arc AB s r Θ
rΘ
10 2 r
r2 Θ
1
2
4
5
rΘ
2
10 r Θ
2
r
r rΘ
rΘ
8
rΘ
2
10 r Θ
8
29) If the roots of x2
r1 3 and r2 3 roots of x2
r1
r2
r1 3
r1 r2
3
3
r1 3
r2 3
2
rΘ 8 rΘ 2
16
0
23
x 2 0
r1 r2
ax b 0
r1 3 r2 3
3 r1 r2 3
r2 3
1
1
r1 3
r2 3
x 2 0, find a and b.
2 and r1
r1 3
b and
2, 8
rΘ
ax b 0 are the cubes of the roots of x2
Let r1 and r2 be the roots of x2
r1 3 r2 3
rΘ
2
5
1
r2
r2 3 a
8
3 r1 2 r2
3 r1 r2 r1
r2
1 3 2
1
5
5, 8
a, b
30) How many positive integers x have the property that 14 is the remainder when
2014 is divided by x ?
2014 qx 14
4
2000 2 5
qx 2000 so x divides 2000 and x 14.
3
4 1 3 1
20 divisors. Of these 1, 2, 4, 8, 5, 10 are less than 14.
The number of x satisfying the given conditions is 20 6 14
31) Find the smallest positive value of x (in radians) such that tan 2 x
tan 2 x
cos x
cos x
cos x
cos x
sin 2 x
cos 2 x
1 sin 2 x
cos 2 x
1
sin x
sin x
sin x
sin x
sin 2 x
cos 2 x
sin 2 x
cos x
cos x
sin x
sin x
sin 2 x
cos x
cos x
sin x
sin x
cos2 x
2 sin 2 x
1
2 sin x cos x sin2 x
cos2 x sin2 x
sin 2 x
1
2
2x
Π
6
x
Π
12
.
B
32 How many paths are there from A to B,
if at each intersection you can only move
in the indicated direction s ?
90
1
89
1
1
6
82
4
A1
16
2
60
6
22
22
Number of paths = 90
33) If x2
x
2
6 x 6 0 , what is the value of x3
6x 6 0
x
2
7 x2
2014 xx2
x3
7 x2
2014 x 6 x
3
2
7x
6 x x 2014
2014
2014 6 2014 2020
20 n
34) For how many integers n is
20 n
14 n
2014?
6x 6
x3
x
7 x2
an integer?
14 n
6
1 14 n
14 n divides 6
14 n
1, 2, 3, 6
Thus 8 values of 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 N ?
M N
10500
3
M N
2
9
1
2
10500
3
101000
M N 2
2
9
10500
1 2 10500
2
101000 1
9
M N 2 111
M N 222
1
2
1
1
10500 1
9
1 2 111
2 – 444
222
222
– 444
2222
4444
222
217
7778
4
1
1000 2s minus 500 4s
From left to right: 499 twos, 1 one, 499 sevens and 1 eight.
499 2
1 499 7
8 499 2 7
9 499 9
9 500 9
4500
6
36) What is the smallest integer that is greater than
6
6
5
4
5
2
3
3
?
3
2
4
5
6
5
3
6
5
6
6 5 3
15 5 3
2
20 5 3
3
15 5 3
4
6 5 3
5
3
6
5
3
6
5
6
6 5 3
15 5 3
2
20 5 3
3
15 5 3
4
6 5 3
5
3
6
5
3
5
6
5
5
3
6
2 5
6
5
3
5
3
4
4
15 5 3
6
4
0
3
2
2
15 5 3
2
2
0
4
0
3
6
6
5
3
5
3
2 125 12 25 3 15 5 9
6
27
6
2 125 1125 675 27
6
Since
3
5
3
2 1952
3904
6
1
5
3
3904
37) The geometric mean of a set of k positive real numbers x1 , x2 , x3 ,
, xk is
1k
x1 x2
xk . Find the positive integer n such that the geometric mean of the
set of all positive integer divisors of n is 70.
If n has prime factorization n p1 Α1 p2 Α2
k Α1 1
Α2 1
Αj 1 .
If all of the Αi are even, k is odd and
divisor
n . Then the product
k 1
2
x1 x2
pj Αj , the number of divisors of n is
of the divisors of n appear in pairs xi and
xk
nk
1 2
n1 2
n
k2
and x1 x2
xk
If not all of the Αi are even, k is even and the divisors of n appear in pairs xi and
Then the product
x1 x2
Thus n1 2
n 4900
70
xk
nk 2 and x1 x2
xk
1k
n
x1
1k
n
x1
n1 2 .
38 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
The area of an equilateral triangle of side length S is A
s2
4
3 . Assuming that the
original triangle has side lenght S,
The first shaded triangle has side length
S
2
and A1
The three shaded triangles in A2 have side length
S
4
2
S 2
3
4
and A2
4
S 2 3
3
4
with the remaining
n1 2 .
The nine shaded triangles in A3 have side length
S
8
and A3
8
S 2 32
3
4
Continuing this pattern:
"
! An
n 1
"
! An
n 1
"
! An
n 1
2
3
4
S2
42
3 1
S2
42
3
S 2
4
S 2 3
s2
8
S 2 32
3
4
Using the sum of the geormetric series with r
32
24
22
1
1
With s 1
3
3
4
3
4
3
4
3
4
3
An
4
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.
0
0
0
0
0
0
0
6
5
4
3
2
1
0
12
10
8
6
4
2
0
15
12
9
0
6
3
0
21
17
13
3
8
4
0
27
22
17
12
10
5
0
75
66
51
24
30
15
Labeling each node with the number of rectangles with the node as lower left
corner and adding:
75 66 51 24 30 15 261
40 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 3 square meters, what is the area
of the gravel walkway ?
2
2
2
2
2
S2
2
S1
2 3
4
30
2
Let the sides of the inside and outside triantles be S1 and S2 respectively.
The areas are respectively
Aw
3
4
S2
3
4
2
3
4
S1 2 and
3
4
S2 2 . Hence the area of the walkway is
S1 2 .
From the diagram above we can see that S2
S1
4
3.
2
S1 4
3
Aw
4
Aw
3 2
3
3 S1
S1
4
2
3
3
4
S1
2
3 2
3 200
3 S1
48
S1
2
12
Using the given area of the inside triangle:
Aw
8
3
12
3
4
S1
3 1200 12
2
30, 000
1212
3
3
S1
200
3
41 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.
s
10
s
L3
10
L2
s
L1
10
L1 =
10 2
10 2 =
s
10
2
L2 =
10
10
=
2s
10
2
Lk =
10
10
=
ks
10
2
s
2s 2
ks 2
9
T=2
k
s=
2
1
s
2s 2
ks 2
Lk + L10 = 2
T = 20
2
s
10
k = 1, 2, 3,
9 10
2
1 = 10
10
2 s