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Purple Comet Problems
2011 Purple Comet Problems
2011 Purple Comet Problems
Part of
Purple Comet Problems
Subcontests
(30)
28
1
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Purple Comet 2011 HS Problem 28
Pictured below is part of a large circle with radius
30
30
30
. There is a chain of three circles with radius
3
3
3
, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius
2
2
2
each tangent to two of the radius
3
3
3
circles. The distance between the centers of the two circles with radius
2
2
2
can be written as
a
b
−
c
d
\textstyle\frac{a\sqrt b-c}d
d
a
b
−
c
, where
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
and
d
d
d
are positive integers,
c
c
c
and
d
d
d
are relatively prime, and
b
b
b
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.[asy] size(200); defaultpen(linewidth(0.5)); real r=aCos(79/81); pair x=dir(270+r)*27,y=dir(270-r)*27; draw(arc(origin,30,210,330)); draw(circle(x,3)^^circle(y,3)^^circle((0,-27),3)); path arcl=arc(y,5,0,180), arcc=arc((0,-27),5,0,180), arcr=arc(x,5,0,180); pair centl=intersectionpoint(arcl,arcc), centr=intersectionpoint(arcc,arcr); draw(circle(centl,2)^^circle(centr,2)); dot(x^^y^^(0,-27)^^centl^^centr,linewidth(2)); [/asy]
24
1
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Purple Comet 2011 HS Problem 24
The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers
m
m
m
,
n
n
n
, and
p
p
p
such that the ratio of the area of the hexagon to the area of the square can be written as
m
+
n
p
\tfrac{m+\sqrt{n}}{p}
p
m
+
n
where
m
m
m
and
p
p
p
are relatively prime. Find
m
+
n
+
p
m + n + p
m
+
n
+
p
.[asy] import graph; size(4cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); filldraw((1.13,2.5)--(-0.13,2.5)--(-0.13,1.23)--(1.13,1.23)--cycle,grey); draw((0,1)--(1,1)); draw((1,1)--(1.5,1.87)); draw((1.5,1.87)--(1,2.73)); draw((1,2.73)--(0,2.73)); draw((0,2.73)--(-0.5,1.87)); draw((-0.5,1.87)--(0,1)); draw((1.13,2.5)--(-0.13,2.5)); draw((-0.13,2.5)--(-0.13,1.23)); draw((-0.13,1.23)--(1.13,1.23)); draw((1.13,1.23)--(1.13,2.5)); [/asy]
26
1
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Purple Comet 2011 HS Problem 26
The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of
16
16
16
points and have area of at least six square units? [asy] import graph; size(7cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; dot((-2,4),linewidth(6pt) + dotstyle); dot((-1,4),linewidth(6pt) + dotstyle); dot((0,4),linewidth(6pt) + dotstyle); dot((1,4),linewidth(6pt) + dotstyle); dot((2,4),linewidth(6pt) + dotstyle); dot((3,4),linewidth(6pt) + dotstyle); dot((4,4),linewidth(6pt) + dotstyle); dot((-3,2),linewidth(6pt) + dotstyle); dot((-2,2),linewidth(6pt) + dotstyle); dot((-1,2),linewidth(6pt) + dotstyle); dot((0,2),linewidth(6pt) + dotstyle); dot((1,2),linewidth(6pt) + dotstyle); dot((2,2),linewidth(6pt) + dotstyle); dot((3,2),linewidth(6pt) + dotstyle); dot((4,2),linewidth(6pt) + dotstyle); dot((5,2),linewidth(6pt) + dotstyle); [/asy]
22
1
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Purple Comet 2011 HS Problem 22
Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle (shaded in the diagram below) congruent to the other fi ve is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other fi ve circles, then it will turn through an angle of
k
k
k
degrees before it returns to its starting position. Find
k
k
k
.[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((2.96,2.58), 1),grey); draw(circle((-1,3), 1)); draw(circle((1,3), 1)); draw(circle((1.62,1.1), 1)); draw(circle((0,-0.08), 1)); draw(circle((-1.62,1.1), 1)); [/asy]
19
2
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Purple Comet 2011 HS Problem 19
The diagrams below shows a
2
2
2
by
2
2
2
grid made up of four
1
1
1
by
1
1
1
squares. Shown are two paths along the grid from the lower left corner to the upper right corner of the grid, one with length
4
4
4
and one with length
6
6
6
. A path may not intersect itself by moving to a point where the path has already been. Find the sum of the lengths of all the paths from the lower left corner to the upper right corner of the grid. [asy] import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,4)--(-1,2), linewidth(1.6)); draw((-1,4)--(1,4), linewidth(1.6)); draw((1,4)--(1,2), linewidth(1.6)); draw((-1,2)--(1,2), linewidth(1.6)); draw((0,4)--(0,2), linewidth(1.6)); draw((-1,3)--(1,3), linewidth(1.6)); draw((-0.5,1)--(-2.5,1), linewidth(1.6)); draw((-2.5,1)--(-2.5,-1), linewidth(1.6)); draw((-1.5,1)--(-1.5,-1), linewidth(1.6)); draw((-2.5,0)--(-0.5,0), linewidth(1.6)); draw((0.5,1)--(0.5,-1), linewidth(1.6)); draw((2.5,1)--(2.5,-1), linewidth(1.6)); draw((0.5,-1)--(2.5,-1), linewidth(1.6)); draw((1.5,1)--(1.5,-1), linewidth(1.6)); draw((0.5,0)--(2.5,0), linewidth(1.6)); draw((0.5,1)--(0.5,0), linewidth(4) + red); draw((0.5,0)--(1.5,0), linewidth(4) + red); draw((0.5,-1)--(1.5,-1), linewidth(4) + red); draw((1.5,0)--(1.5,-1), linewidth(4) + red); draw((0.5,1)--(2.5,1), linewidth(4) + red); draw((-0.5,1)--(-0.5,-1), linewidth(4) + red); draw((-2.5,-1)--(-0.5,-1), linewidth(4) + red); [/asy]
Purple Comet 2011 MS Problem 19
How many ordered pairs of sets
(
A
,
B
)
(A, B)
(
A
,
B
)
have the properties:1. A\subseteq \{1, 2, 3, 4, 5, 6\} 2.
B
⊆
{
2
,
3
,
4
,
5
,
6
,
7
,
8
}
B\subseteq\{2, 3, 4, 5, 6, 7, 8\}
B
⊆
{
2
,
3
,
4
,
5
,
6
,
7
,
8
}
3.
A
∩
B
A\cap B
A
∩
B
has exactly
3
3
3
elements.
30
1
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Purple Comet 2011 HS Problem 30
Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere,
R
R
R
, contains the four congruent spheres so that all four are internally tangent to
R
R
R
. A smaller sphere,
S
S
S
, sits in the space between the four congruent spheres so that all four are externally tangent to
S
S
S
. The ratio of the surface area of
R
R
R
to the surface area of
S
S
S
can be written
m
+
n
m+\sqrt{n}
m
+
n
where
m
m
m
and
n
n
n
are positive integers. Find
m
+
n
m + n
m
+
n
.
29
1
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Purple Comet 2011 HS Problem 29
Let
S
S
S
be a randomly selected four-element subset of
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
\{1, 2, 3, 4, 5, 6, 7, 8\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
. Let
m
m
m
and
n
n
n
be relatively prime positive integers so that the expected value of the maximum element in
S
S
S
is
m
n
\dfrac{m}{n}
n
m
. Find
m
+
n
m + n
m
+
n
.
27
1
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Purple Comet 2011 HS Problem 27
Find the smallest prime number that does not divide
9
+
9
2
+
9
3
+
⋯
+
9
2010
.
9+9^2+9^3+\cdots+9^{2010}.
9
+
9
2
+
9
3
+
⋯
+
9
2010
.
25
1
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Purple Comet 2011 HS Problem 25
Find the remainder when
A
=
3
3
⋅
3
3
33
⋅
33
3
333
⋅
333
3
3333
A=3^3\cdot 33^{33}\cdot 333^{333}\cdot 3333^{3333}
A
=
3
3
⋅
3
3
33
⋅
33
3
333
⋅
333
3
3333
is divided by
100
100
100
.
23
1
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Purple Comet 2011 HS Problem 23
Let
x
x
x
be a real number in the interval
(
0
,
π
2
)
\left(0,\dfrac{\pi}{2}\right)
(
0
,
2
π
)
such that
1
sin
x
cos
x
+
2
cot
2
x
=
1
2
\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}
sin
x
cos
x
1
+
2
cot
2
x
=
2
1
. Evaluate
1
sin
x
cos
x
−
2
cot
2
x
\dfrac{1}{\sin x \cos x}-2\cot 2x
sin
x
cos
x
1
−
2
cot
2
x
.
21
1
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Purple Comet 2011 HS Problem 21
If a, b, and c are non-negative real numbers satisfying
a
+
b
+
c
=
400
a + b + c = 400
a
+
b
+
c
=
400
, fi nd the maximum possible value of
2
a
+
b
+
2
b
+
c
+
2
c
+
a
\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}
2
a
+
b
+
2
b
+
c
+
2
c
+
a
.
15
2
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Purple Comet 2011 HS Problem 15
A pyramid has a base which is an equilateral triangle with side length
300
300
300
centimeters. The vertex of the pyramid is
100
100
100
centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of
134
134
134
centimeters, how many centimeters above the base of the pyramid is the mouse?
Purple Comet 2011 MS Problem 15
In the diagram below,
A
B
‾
\overline{AB}
A
B
and
C
D
‾
\overline{CD}
C
D
are parallel, \angle BXY = 45^\circ,
∠
D
Z
Y
=
2
5
∘
\angle DZY = 25^\circ
∠
D
Z
Y
=
2
5
∘
, and
X
Y
=
Y
Z
XY = YZ
X
Y
=
Y
Z
. What is the degree measure of
∠
Y
X
Z
\angle YXZ
∠
Y
XZ
? [asy] import graph; usepackage("amsmath"); size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-2,4)--(3,4)); draw((-2,2)--(3,2)); draw((0,4)--(1,3)); draw((1,3)--(-1.14,2)); label("
A
A
A
",(-2.13,4.6),SE*labelscalefactor); label("
B
B
B
",(2.8,4.6),SE*labelscalefactor); label("
C
C
C
",(-2.29,1.8),SE*labelscalefactor); label("
D
D
D
",(2.83,1.8),SE*labelscalefactor); label("
4
5
∘
45^\circ
4
5
∘
",(0.49,3.9),SE*labelscalefactor); label("
2
5
∘
25^\circ
2
5
∘
",(-0.26,2.4),SE*labelscalefactor); label("
Y
Y
Y
",(1.21,3.2),SE*labelscalefactor); label("
X
X
X
",(-0.16,4.6),SE*labelscalefactor); label("
Z
Z
Z
",(-1.28,1.8),SE*labelscalefactor); dot((-2,4),dotstyle); dot((3,4),dotstyle); dot((-2,2),dotstyle); dot((3,2),dotstyle); dot((0,4),dotstyle); dot((1,3),dotstyle); dot((-1.14,2),dotstyle); [/asy]
10
2
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Purple Comet 2011 HS Problem 10
The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is
m
n
\dfrac{m}{n}
n
m
where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.[asy] size(150); defaultpen(linewidth(0.8)); filldraw(circle((0,0.5),.5),gray); filldraw(circle((0,-0.5),.5),gray); filldraw(circle((2/3,0),1/3),gray); filldraw(circle((-2/3,0),1/3),gray); draw(unitcircle); [/asy]
Purple Comet 2011 MS Problem 10
Five rays
O
A
→
\overrightarrow{OA}
O
A
,
O
B
→
\overrightarrow{OB}
OB
,
O
C
→
\overrightarrow{OC}
OC
,
O
D
→
\overrightarrow{OD}
O
D
, and
O
E
→
\overrightarrow{OE}
OE
radiate in a clockwise order from
O
O
O
forming four non-overlapping angles such that
∠
E
O
D
=
2
∠
C
O
B
\angle EOD = 2\angle COB
∠
EO
D
=
2∠
COB
,
∠
C
O
B
=
2
∠
B
O
A
\angle COB = 2\angle BOA
∠
COB
=
2∠
BO
A
, while
∠
D
O
C
=
3
∠
B
O
A
\angle DOC = 3\angle BOA
∠
D
OC
=
3∠
BO
A
. If
E
E
E
,
O
O
O
,
A
A
A
are collinear with
O
O
O
between
A
A
A
and
E
E
E
, what is the degree measure of
∠
D
O
B
?
\angle DOB?
∠
D
OB
?
16
2
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Purple Comet 2011 HS Problem 16
Evaluate
1
3
−
2
3
+
3
3
−
4
3
+
5
3
−
⋯
+
10
1
3
1^3-2^3+3^3-4^3+5^3-\cdots+101^3
1
3
−
2
3
+
3
3
−
4
3
+
5
3
−
⋯
+
10
1
3
.
Purple Comet 2011 MS Problem 16
Let
a
a
a
and
b
b
b
be nonzero real numbers such that
1
3
a
+
1
b
=
2011
\tfrac{1}{3a}+\tfrac{1}{b}=2011
3
a
1
+
b
1
=
2011
and
1
a
+
1
3
b
=
1
\tfrac{1}{a}+\tfrac{1}{3b}=1
a
1
+
3
b
1
=
1
. What is the quotient when
a
+
b
a+b
a
+
b
is divided by
a
b
ab
ab
?
13
2
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Purple Comet 2011 HS Problem 13
A
3
3
3
by
3
3
3
determinant has three entries equal to
2
2
2
, three entries equal to
5
5
5
, and three entries equal to
8
8
8
. Find the maximum possible value of the determinant.
Purple Comet 2011 MS Problem 13
The diagram shows two equilateral triangles with side length
4
4
4
mounted on two adjacent sides of a square also with side length
4
4
4
. The distance between the two vertices marked
A
A
A
and
B
B
B
can be written as
m
+
n
\sqrt{m}+\sqrt{n}
m
+
n
for two positive integers
m
m
m
and
n
n
n
. Find
m
+
n
m + n
m
+
n
.[asy] size(120); defaultpen(linewidth(0.7)+fontsize(11pt)); draw(unitsquare); draw((0,1)--(1/2,1+sqrt(3)/2)--(1,1)--(1+sqrt(3)/2,1/2)--(1,0)); label("
A
A
A
",(1/2,1+sqrt(3)/2),N); label("
B
B
B
",(1+sqrt(3)/2,1/2),E); [/asy]
2
2
Hide problems
Purple Comet 2011 HS Problem 2
The target below is made up of concentric circles with diameters
4
4
4
,
8
8
8
,
12
12
12
,
16
16
16
, and
20
20
20
. The area of the dark region is
n
π
n\pi
nπ
. Find
n
n
n
.[asy] size(150); defaultpen(linewidth(0.8)); int i; for(i=5;i>=1;i=i-1) { if (floor(i/2)==i/2) { filldraw(circle(origin,4*i),white); } else { filldraw(circle(origin,4*i),red); } } [/asy]
Purple Comet 2011 MS Problem 2
The diagram below shows a
12
12
12
-sided figure made up of three congruent squares. The figure has total perimeter
60
60
60
. Find its area.[asy] size(150); defaultpen(linewidth(0.8)); path square=unitsquare; draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square); [/asy]
17
2
Hide problems
Purple Comet 2011 HS Problem 17
In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?
Purple Comet 2011 MS Problem 17
Find the number of ordered quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
where each of
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
are (not necessarily distinct) elements of
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
\{1, 2, 3, 4, 5, 6, 7\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
and
3
a
b
c
+
4
a
b
d
+
5
b
c
d
3abc + 4abd + 5bcd
3
ab
c
+
4
ab
d
+
5
b
c
d
is even. For example,
(
2
,
2
,
5
,
1
)
(2, 2, 5, 1)
(
2
,
2
,
5
,
1
)
and
(
3
,
1
,
4
,
6
)
(3, 1, 4, 6)
(
3
,
1
,
4
,
6
)
satisfy the conditions.
18
2
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Purple Comet 2011 HS Problem 18
Let
a
a
a
be a positive real number such that
a
2
a
4
−
a
2
+
1
=
4
37
\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}
a
4
−
a
2
+
1
a
2
=
37
4
. Then
a
3
a
6
−
a
3
+
1
=
m
n
\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}
a
6
−
a
3
+
1
a
3
=
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
.
Purple Comet 2011 MS Problem 18
Find the positive integer
n
n
n
so that
n
2
n^2
n
2
is the perfect square closest to
8
+
16
+
24
+
⋯
+
8040.
8 + 16 + 24 + \cdots + 8040.
8
+
16
+
24
+
⋯
+
8040.
20
2
Hide problems
Purple Comet 2011 HS Problem 20
Points
A
A
A
and
B
B
B
are the endpoints of a diameter of a circle with center
C
C
C
. Points
D
D
D
and
E
E
E
lie on the same diameter so that
C
C
C
bisects segment
D
E
‾
\overline{DE}
D
E
. Let
F
F
F
be a randomly chosen point within the circle. The probability that
△
D
E
F
\triangle DEF
△
D
EF
has a perimeter less than the length of the diameter of the circle is
17
128
\tfrac{17}{128}
128
17
. There are relatively prime positive integers m and n so that the ratio of
D
E
DE
D
E
to
A
B
AB
A
B
is
m
n
.
\tfrac{m}{n}.
n
m
.
Find
m
+
n
m + n
m
+
n
.
Purple Comet 2011 MS Problem 20
Let
V
V
V
be the set of vertices of a regular
25
25
25
sided polygon with center at point
C
.
C.
C
.
How many triangles have vertices in
V
V
V
and contain the point
C
C
C
in the interior of the triangle?
14
2
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Purple Comet 2011 HS Problem 14
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is
m
+
n
k
\tfrac{m+\sqrt{n}}{k}
k
m
+
n
where
m
m
m
,
n
n
n
, and
k
k
k
are integers, and
k
k
k
is not divisible by the square of any prime. Find
m
+
n
+
k
m + n + k
m
+
n
+
k
.
Purple Comet 2011 MS Problem 14
The five-digit number
12110
12110
12110
is divisible by the sum of its digits
1
+
2
+
1
+
1
+
0
=
5.
1 + 2 + 1 + 1 + 0 = 5.
1
+
2
+
1
+
1
+
0
=
5.
Find the greatest five-digit number which is divisible by the sum of its digits
12
2
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Purple Comet 2011 MS Problem 12
Find the area of the region in the coordinate plane satisfying the three conditions
⋆
\star
⋆
x \le 2y
⋆
\star
⋆
y \le 2x
⋆
\star
⋆
x + y \le 60.
Purple Comet 2011 MS Problem 12
When Troy writes his digits, his
0
0
0
,
1
1
1
, and
8
8
8
look the same right-side-up and upside-down as seen in the figure below. His
6
6
6
and
9
9
9
look like upside-down images of each other. None of his other digits look like digits when they are inverted. How many diff erent five-digit numbers (which do not begin with the digit zero) can Troy write which read the same right-side-up and upside-down?[asy] frame l; label(l,"\textsf{0}\qquad \textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}"); add(rotate(180)*l); label("\textsf{0}\qquad\textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}",(0,20)); [/asy]
11
2
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Purple Comet 2011 HS Problem 11
Six distinct positive integers are randomly chosen between
1
1
1
and
2011
;
2011;
2011
;
inclusive. The probability that some pair of the six chosen integers has a di fference that is a multiple of
5
5
5
is
n
n
n
percent. Find
n
.
n.
n
.
Purple Comet 2011 MS Problem 11
How many numbers are there that appear both in the arithmetic sequence
10
,
16
,
22
,
28
,
.
.
.
1000
10, 16, 22, 28, ... 1000
10
,
16
,
22
,
28
,
...1000
and the arithmetic sequence
10
,
21
,
32
,
43
,
.
.
.
,
1000
?
10, 21, 32, 43, ..., 1000?
10
,
21
,
32
,
43
,
...
,
1000
?
9
2
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Purple Comet 2011 HS Problem 9
There are integers
m
m
m
and
n
n
n
so that
9
+
11
9 +\sqrt{11}
9
+
11
is a root of the polynomial
x
2
+
m
x
+
n
.
x^2 + mx + n.
x
2
+
m
x
+
n
.
Find
m
+
n
.
m + n.
m
+
n
.
Purple Comet 2011 MS Problem 9
A jar contains one quarter red jelly beans and three quarters blue jelly beans. If Chris removes three quarters of the red jelly beans and one quarter of the blue jelly beans, what percent of the jelly beans remaining in the jar will be red?
8
2
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Purple Comet 2011 HS Problem 8
When
126
126
126
is added to its reversal,
621
,
621,
621
,
the sum is
126
+
621
=
747.
126 + 621 = 747.
126
+
621
=
747.
Find the greatest integer which when added to its reversal yields
1211.
1211.
1211.
Purple Comet 2011 MS Problem 8
A square measuring
15
15
15
by
15
15
15
is partitioned into five rows of fi ve congruent squares as shown below. The small squares are alternately colored black and white as shown. Find the total area of the part colored black.[asy] size(150); defaultpen(linewidth(0.8)); int i,j; for(i=1;i<=5;i=i+1) { for(j=1;j<=5;j=j+1) { if (floor((i+j)/2)==((i+j)/2)) { filldraw(shift((i-1,j-1))*unitsquare,gray); } else { draw(shift((i-1,j-1))*unitsquare); } } } [/asy]
7
2
Hide problems
Purple Comet 2011 MS Problem 7
Find the prime number
p
p
p
such that
71
p
+
1
71p + 1
71
p
+
1
is a perfect square.
Purple Comet 2011 MS Problem 7
When
12
1
8
12{}^1{}^8
12
1
8
is divided by
18
1
2
18{}^1{}^2
18
1
2
, the result is
(
m
n
)
3
(\tfrac{m}{n})^3
(
n
m
)
3
, where
m
m
m
and
n
n
n
are relatively prime integers. Find
m
−
n
m-n
m
−
n
.
6
2
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Purple Comet 2011 HS Problem 6
Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them
m
n
\frac{m}{n}
n
m
days where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
.
m + n.
m
+
n
.
Purple Comet 2011 MS Problem 6
The following addition problem is not correct if the numbers are interpreted as base 10 numbers. In what number base is the problem correct?
66
+
87
+
85
+
48
=
132
66+ 87+ 85 +48 = 132
66
+
87
+
85
+
48
=
132
5
2
Hide problems
Purple Comet 2011 HS Problem 5
Let
a
1
=
2
,
a_1 = 2,
a
1
=
2
,
and for
n
≥
1
,
n\ge 1,
n
≥
1
,
let
a
n
+
1
=
2
a
n
+
1.
a_{n+1} = 2a_n + 1.
a
n
+
1
=
2
a
n
+
1.
Find the smallest value of an
a
n
a_n
a
n
that is not a prime number.
Purple Comet 2011 MS Problem 5
Given that:
6
11
−
10
19
=
9
19
−
n
11
\tfrac{6}{11}-\tfrac{10}{19}=\tfrac{9}{19}-\tfrac{n}{11}
11
6
−
19
10
=
19
9
−
11
n
, find
n
n
n
.
4
2
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Purple Comet 2011 HS Problem 4
Five non-overlapping equilateral triangles meet at a common vertex so that the angles between adjacent triangles are all congruent. What is the degree measure of the angle between two adjacent triangles?[asy] size(100); defaultpen(linewidth(0.7)); path equi=dir(300)--dir(240)--origin--cycle; for(int i=0;i<=4;i=i+1) draw(rotate(72*i,origin)*equi); [/asy]
Purple Comet 2011 MS Problem 4
Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in the bottle?
3
2
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Purple Comet 2011 MS Problem 3
Shirley went to the store planning to buy
120
120
120
balloons for
10
10
10
dollars. When she arrived, she was surprised to nd that the balloons were on sale for
20
20
20
percent less than expected. How many balloons could Shirley buy for her
10
10
10
dollars?
Purple Comet 2011 MS Problem 3
Find the sum of all two-digit integers which are both prime and are 1 more than a multiple of 10.
1
2
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Purple Comet 2011 HS Problem 1
The ratio of
3
3
3
to the positive number
n
n
n
is the same as the ratio of
n
n
n
to
192.
192.
192.
Find
n
.
n.
n
.
Purple Comet 2011 MS Problem 1
There are relatively prime positive integers
m
m
m
and
n
n
n
so that
1
2
1
3
1
4
+
1
4
+
1
4
+
1
3
1
4
+
1
4
+
1
4
=
m
n
.
\dfrac{\dfrac{1}{2}}{\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}+\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}}=\dfrac{m}{n}.
4
1
+
4
1
+
4
1
3
1
+
4
1
+
4
1
+
4
1
3
1
2
1
=
n
m
.
Find
m
+
2
n
m+2n
m
+
2
n
.