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MMATHS problems
2018 MMATHS
2018 MMATHS
Part of
MMATHS problems
Subcontests
(6)
Mixer Round
1
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2018 MMATHS Mixer Round - Math Majors of America Tournament for High Schools
p1. Suppose
x
y
=
0.
a
b
‾
\frac{x}{y} = 0.\overline{ab}
y
x
=
0.
ab
where
x
x
x
and
y
y
y
are relatively prime positive integers and
a
b
+
a
+
b
+
1
ab + a + b + 1
ab
+
a
+
b
+
1
is a multiple of
12
12
12
. Find the sum of all possible values of
y
y
y
. p2. Let
A
A
A
be the set of points
{
(
0
,
0
)
,
(
2
,
0
)
,
(
0
,
2
)
,
(
2
,
2
)
,
(
3
,
1
)
,
(
1
,
3
)
}
\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}
{(
0
,
0
)
,
(
2
,
0
)
,
(
0
,
2
)
,
(
2
,
2
)
,
(
3
,
1
)
,
(
1
,
3
)}
. How many distinct circles pass through at least three points in
A
A
A
? p3. Jack and Jill need to bring pails of water home. The river is the
x
x
x
-axis, Jack is initially at the point
(
−
5
,
3
)
(-5, 3)
(
−
5
,
3
)
, Jill is initially at the point
(
6
,
1
)
(6, 1)
(
6
,
1
)
, and their home is at the point
(
0
,
h
)
(0, h)
(
0
,
h
)
where
h
>
0
h > 0
h
>
0
. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is
h
h
h
? p4. What is the largest perfect square which is not a multiple of
10
10
10
and which remains a perfect square if the ones and tens digits are replaced with zeroes? p5. In convex polygon
P
P
P
, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides
P
P
P
could have? p6. How many polynomials
p
(
x
)
p(x)
p
(
x
)
of degree exactly
3
3
3
with real coefficients satisfy
p
(
0
)
,
p
(
1
)
,
p
(
2
)
,
p
(
3
)
∈
{
0
,
1
,
2
}
?
p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?
p
(
0
)
,
p
(
1
)
,
p
(
2
)
,
p
(
3
)
∈
{
0
,
1
,
2
}?
p7. Six spheres, each with radius
4
4
4
, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius
13
13
13
, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere? p8. You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? p9. Quadrilateral
A
B
C
D
ABCD
A
BC
D
has
A
B
‾
=
40
\overline{AB} = 40
A
B
=
40
,
C
D
‾
=
10
\overline{CD} = 10
C
D
=
10
,
A
D
‾
=
B
C
‾
\overline{AD} = \overline{BC}
A
D
=
BC
,
m
∠
B
A
D
=
2
0
o
m\angle BAD = 20^o
m
∠
B
A
D
=
2
0
o
, and
m
∠
A
B
C
=
7
0
o
m \angle ABC = 70^o
m
∠
A
BC
=
7
0
o
. What is the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
? p10. We say that a permutation
σ
\sigma
σ
of the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,..., n\}
{
1
,
2
,
...
,
n
}
preserves divisibilty if
σ
(
a
)
\sigma (a)
σ
(
a
)
divides
σ
(
b
)
\sigma (b)
σ
(
b
)
whenever
a
a
a
divides
b
b
b
. How many permutations of
{
1
,
2
,
.
.
.
,
40
}
\{1, 2,..., 40\}
{
1
,
2
,
...
,
40
}
preserve divisibility? (A permutation of
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,..., n\}
{
1
,
2
,
...
,
n
}
is a function
σ
\sigma
σ
from
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,..., n\}
{
1
,
2
,
...
,
n
}
to itself such that for any
b
∈
{
1
,
2
,
.
.
.
,
n
}
b \in \{1, 2,..., n\}
b
∈
{
1
,
2
,
...
,
n
}
, there exists some
a
∈
{
1
,
2
,
.
.
.
,
n
}
a \in \{1, 2,..., n\}
a
∈
{
1
,
2
,
...
,
n
}
satisfying
σ
(
a
)
=
b
\sigma (a) = b
σ
(
a
)
=
b
.) p11. In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected? https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.pngp12. Let
S
S
S
be the set of the
125
125
125
points in three-dimension space of the form
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
where
x
x
x
,
y
y
y
, and
z
z
z
are integers between
1
1
1
and
5
5
5
, inclusive. A family of snakes lives at the point
(
1
,
1
,
1
)
(1, 1, 1)
(
1
,
1
,
1
)
, and one day they decide to move to the point
(
5
,
5
,
5
)
(5, 5, 5)
(
5
,
5
,
5
)
. Snakes may slither only in increments of
(
1
,
0
,
0
)
(1,0,0)
(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
(0, 1, 0)
(
0
,
1
,
0
)
, and
(
0
,
0
,
1
)
(0, 0, 1)
(
0
,
0
,
1
)
. Given that at least one snake has slithered through each point of
S
S
S
by the time the entire family has reached
(
5
,
5
,
5
)
(5, 5, 5)
(
5
,
5
,
5
)
, what is the smallest number of snakes that could be in the family? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
4
1
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2018 MMATHS Tiebreaker p4 - s_n = cs_{n-1} + ds_{n-2}, t_n=s_n mod 2018
A sequence of integers fsng is defined as follows: fix integers
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
, then set
s
1
=
a
s_1 = a
s
1
=
a
,
s
2
=
b
s_2 = b
s
2
=
b
, and
s
n
=
c
s
n
−
1
+
d
s
n
−
2
s_n = cs_{n-1} + ds_{n-2}
s
n
=
c
s
n
−
1
+
d
s
n
−
2
for all
n
≥
3
n \ge 3
n
≥
3
. Create a second sequence
{
t
n
}
\{t_n\}
{
t
n
}
by defining each
t
n
t_n
t
n
to be the remainder when
s
n
s_n
s
n
is divided by
2018
2018
2018
(so we always have
0
≤
t
n
≤
2017
0 \le t_n \le 2017
0
≤
t
n
≤
2017
). Let
N
=
(
201
8
2
)
!
N = (2018^2)!
N
=
(
201
8
2
)!
. Prove that
t
N
=
t
2
N
t_N = t_{2N}
t
N
=
t
2
N
regardless of the choices of
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
.
3
1
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2018 MMATHS Tiebreaker p3 - center of circle in interior of cyclic n-gon
Suppose
n
n
n
points are uniformly chosen at random on the circumference of the unit circle and that they are then connected with line segments to form an
n
n
n
-gon. What is the probability that the origin is contained in the interior of this
n
n
n
-gon? Give your answer in terms of
n
n
n
, and consider only
n
≥
3
n \ge 3
n
≥
3
.
2
1
Hide problems
2018 MMATHS Tiebreaker p2 - integer triangle with integer area=perimeter
Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.
1
1
Hide problems
2018 MMATHS Tiebreaker p1 - 2 and n tiles in 8x10 grid
Daniel has an unlimited supply of tiles labeled “
2
2
2
” and “
n
n
n
” where
n
n
n
is an integer. Find (with proof) all the values of
n
n
n
that allow Daniel to fill an
8
×
10
8 \times 10
8
×
10
grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by
11
11
11
.
3
Hide problems
2018 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS
Round 5 p13. Circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
, and
ω
3
\omega_3
ω
3
have radii
8
8
8
,
5
5
5
, and
5
5
5
, respectively, and each is externally tangent to the other two. Circle
ω
4
\omega_4
ω
4
is internally tangent to
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
, and
ω
3
\omega_3
ω
3
, and circle
ω
5
\omega_5
ω
5
is externally tangent to the same three circles. Find the product of the radii of
ω
4
\omega_4
ω
4
and
ω
5
\omega_5
ω
5
. p14. Pythagoras has a regular pentagon with area
1
1
1
. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer. p15. Maisy arranges
61
61
61
ordinary yellow tennis balls and
3
3
3
special purple tennis balls into a
4
×
4
×
4
4 \times 4 \times 4
4
×
4
×
4
cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching? Round 6 p16. Points
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
lie on a line (in that order), and
△
B
C
E
\vartriangle BCE
△
BCE
is isosceles with
B
E
‾
=
C
E
‾
\overline{BE} = \overline{CE}
BE
=
CE
. Furthermore,
F
F
F
lies on
B
E
‾
\overline{BE}
BE
and
G
G
G
lies on
C
E
‾
\overline{CE}
CE
such that
△
B
F
D
\vartriangle BFD
△
BF
D
and
△
C
G
A
\vartriangle CGA
△
CG
A
are both congruent to
△
B
C
E
\vartriangle BCE
△
BCE
. Let
H
H
H
be the intersection of
D
F
‾
\overline{DF}
D
F
and
A
G
‾
\overline{AG}
A
G
, and let
I
I
I
be the intersection of
B
E
‾
\overline{BE}
BE
and
A
G
‾
\overline{AG}
A
G
. If
m
∠
B
C
E
=
a
r
c
s
i
n
(
12
13
)
m \angle BCE = arcsin \left( \frac{12}{13} \right)
m
∠
BCE
=
a
rcs
in
(
13
12
)
, what is
H
I
‾
F
I
‾
\frac{\overline{HI}}{\overline{FI}}
F
I
H
I
? p17. Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering. p18. Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, and
e
e
e
be integers satisfying
2
(
2
3
)
2
+
2
3
a
+
2
b
+
(
2
3
)
2
c
+
2
3
d
+
e
=
0
2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0
2
(
3
2
)
2
+
3
2
a
+
2
b
+
(
3
2
)
2
c
+
3
2
d
+
e
=
0
and
25
5
i
+
25
a
−
5
5
i
b
−
5
c
+
5
i
d
+
e
=
0
25\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0
25
5
i
+
25
a
−
5
5
ib
−
5
c
+
5
i
d
+
e
=
0
where
i
=
−
1
i =\sqrt{-1}
i
=
−
1
. Find
∣
a
+
b
+
c
+
d
+
e
∣
|a + b + c + d + e|
∣
a
+
b
+
c
+
d
+
e
∣
. Round 7 p19. What is the greatest number of regions that
100
100
100
ellipses can divide the plane into? Include the unbounded region. p20. All of the faces of the convex polyhedron
P
P
P
are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of
P
P
P
. p21. Find the number of ordered
2018
2018
2018
-tuples of integers
(
x
1
,
x
2
,
.
.
.
.
x
2018
)
(x_1, x_2, .... x_{2018})
(
x
1
,
x
2
,
....
x
2018
)
, where each integer is between
−
201
8
2
-2018^2
−
201
8
2
and
201
8
2
2018^2
201
8
2
(inclusive), satisfying
6
(
1
x
1
+
2
x
2
+
.
.
.
⋅
+
2018
x
2018
)
2
≥
(
2018
)
(
2019
)
(
4037
)
(
x
1
2
+
x
2
2
+
.
.
.
+
x
2018
2
)
.
6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).
6
(
1
x
1
+
2
x
2
+
...
⋅
+
2018
x
2018
)
2
≥
(
2018
)
(
2019
)
(
4037
)
(
x
1
2
+
x
2
2
+
...
+
x
2018
2
)
.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2018 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS
Round 1 p1. Elaine creates a sequence of positive integers
{
s
n
}
\{s_n\}
{
s
n
}
. She starts with
s
1
=
2018
s_1 = 2018
s
1
=
2018
. For
n
≥
2
n \ge 2
n
≥
2
, she sets
s
n
=
1
2
s
n
−
1
s_n =\frac12 s_{n-1}
s
n
=
2
1
s
n
−
1
if
s
n
−
1
s_{n-1}
s
n
−
1
is even and
s
n
=
s
n
−
1
+
1
s_n = s_{n-1} + 1
s
n
=
s
n
−
1
+
1
if
s
n
−
1
s_{n-1}
s
n
−
1
is odd. Find the smallest positive integer
n
n
n
such that
s
n
=
1
s_n = 1
s
n
=
1
, or submit “
0
0
0
” as your answer if no such
n
n
n
exists. p2. Alice rolls a fair six-sided die with the numbers
1
1
1
through
6
6
6
, and Bob rolls a fair eight-sided die with the numbers
1
1
1
through
8
8
8
. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? p3. Four circles each of radius
1
4
\frac14
4
1
are centered at the points
(
±
1
4
,
±
1
4
)
\left( \pm \frac14, \pm \frac14 \right)
(
±
4
1
,
±
4
1
)
, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle? Round 2 p4. If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing? p5. Let
a
1
=
2018
a_1 = 2018
a
1
=
2018
, and for
n
≥
2
n \ge 2
n
≥
2
define
a
n
=
201
8
a
n
−
1
a_n = 2018^{a_{n-1}}
a
n
=
201
8
a
n
−
1
. What is the ones digit of
a
2018
a_{2018}
a
2018
? p6. We can write
(
x
+
35
)
n
=
∑
i
=
0
n
c
i
x
i
(x + 35)^n =\sum_{i=0}^n c_ix^i
(
x
+
35
)
n
=
∑
i
=
0
n
c
i
x
i
for some positive integer
n
n
n
and real numbers
c
i
c_i
c
i
. If
c
0
=
c
2
c_0 = c_2
c
0
=
c
2
, what is
n
n
n
? Round 3 p7. How many positive integers are factors of
12
!
12!
12
!
but not of
(
7
!
)
2
(7!)^2
(
7
!
)
2
? p8. How many ordered pairs
(
f
(
x
)
,
g
(
x
)
)
(f(x), g(x))
(
f
(
x
)
,
g
(
x
))
of polynomials of degree at least
1
1
1
with integer coefficients satisfy
f
(
x
)
g
(
x
)
=
50
x
6
−
3200
f(x)g(x) = 50x^6 - 3200
f
(
x
)
g
(
x
)
=
50
x
6
−
3200
? p9. On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between
1
1
1
and
10
10
10
(inclusive). What is the probability that the average of their three scores is an integer? Round 4 p10. Find the largest positive integer N such that
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
a
−
e
)
(
b
−
c
)
(
b
−
d
)
(
b
−
e
)
(
c
−
d
)
(
c
−
e
)
(
d
−
e
)
(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
a
−
e
)
(
b
−
c
)
(
b
−
d
)
(
b
−
e
)
(
c
−
d
)
(
c
−
e
)
(
d
−
e
)
is divisible by
N
N
N
for all choices of positive integers
a
>
b
>
c
>
d
>
e
a > b > c > d > e
a
>
b
>
c
>
d
>
e
. p11. Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a square pyramid with
A
B
C
D
ABCD
A
BC
D
a square and E the apex of the pyramid. Each side length of
A
B
C
D
E
ABCDE
A
BC
D
E
is
6
6
6
. Let
A
B
C
D
D
′
C
′
B
′
A
′
ABCDD'C'B'A'
A
BC
D
D
′
C
′
B
′
A
′
be a cube, where
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
,
D
D
′
DD'
D
D
′
are edges of the cube. Andy the ant is on the surface of
E
A
B
C
D
D
′
C
′
B
′
A
′
EABCDD'C'B'A'
E
A
BC
D
D
′
C
′
B
′
A
′
at the center of triangle
A
B
E
ABE
A
BE
(call this point
G
G
G
) and wants to crawl on the surface of the cube to
D
′
D'
D
′
. What is the length the shortest path from
G
G
G
to
D
′
D'
D
′
? Write your answer in the form
a
+
b
3
\sqrt{a + b\sqrt3}
a
+
b
3
, where
a
a
a
and
b
b
b
are positive integers. p12. A six-digit palindrome is a positive integer between
100
,
000
100, 000
100
,
000
and
999
,
999
999, 999
999
,
999
(inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2018 MMATHS Individual Round - Math Majors of America Tournament for High School
p1. Five friends arrive at a hotel which has three rooms. Rooms
A
A
A
and
B
B
B
hold two people each, and room
C
C
C
holds one person. How many different ways could the five friends lodge for the night? p2. The set of numbers
{
1
,
3
,
8
,
12
,
x
}
\{1, 3, 8, 12, x\}
{
1
,
3
,
8
,
12
,
x
}
has the same average and median. What is the sum of all possible values of
x
x
x
? (Note that
x
x
x
is not necessarily greater than
12
12
12
.) p3. How many four-digit numbers
A
B
C
D
‾
\overline{ABCD}
A
BC
D
are there such that the three-digit number
B
C
D
‾
\overline{BCD}
BC
D
satisfies
B
C
D
‾
=
1
6
A
B
C
D
‾
\overline{BCD} = \frac16 \overline{ABCD}
BC
D
=
6
1
A
BC
D
? (Note that
A
A
A
must be nonzero.) p4. Find the smallest positive integer
n
n
n
such that
n
n
n
leaves a remainder of
5
5
5
when divided by
14
14
14
,
n
2
n^2
n
2
leaves a remainder of
1
1
1
when divided by
12
12
12
, and
n
3
n^3
n
3
leaves a remainder of
7
7
7
when divided by
10
10
10
. p5. In rectangle
A
B
C
D
ABCD
A
BC
D
, let
E
E
E
lie on
C
D
‾
\overline{CD}
C
D
, and let
F
F
F
be the intersection of
A
C
‾
\overline{AC}
A
C
and
B
E
‾
\overline{BE}
BE
. If the area of
△
A
B
F
\vartriangle ABF
△
A
BF
is
45
45
45
and the area of
△
C
E
F
\vartriangle CEF
△
CEF
is
20
20
20
, find the area of the quadrilateral
A
D
E
F
ADEF
A
D
EF
. p6. If
x
x
x
and
y
y
y
are integers and
14
x
2
y
3
−
38
x
2
+
21
y
3
=
2018
14x^2y^3 - 38x^2 + 21y^3 = 2018
14
x
2
y
3
−
38
x
2
+
21
y
3
=
2018
, what is the value of
x
2
y
x^2y
x
2
y
? p7.
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
all lie on a circle with
A
B
‾
=
B
C
‾
=
C
D
‾
\overline{AB} = \overline{BC} = \overline{CD}
A
B
=
BC
=
C
D
. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral
A
B
C
D
ABCD
A
BC
D
? p8. Compute
∑
m
=
1
∞
∑
n
=
1
∞
m
cos
2
(
n
)
+
n
sin
2
(
m
)
3
m
+
n
(
m
+
n
)
\sum^{\infty}_{m=1} \sum^{\infty}_{n=1} \frac{m\cos^2(n) + n \sin^2(m)}{3^{m+n}(m + n)}
m
=
1
∑
∞
n
=
1
∑
∞
3
m
+
n
(
m
+
n
)
m
cos
2
(
n
)
+
n
sin
2
(
m
)
p9. Diane has a collection of weighted coins with different probabilities of landing on heads, and she flips nine coins sequentially according to a particular set of rules. She uses a coin that always lands on heads for her first and second flips, and she uses a coin that always lands on tails for her third flip. For each subsequent flip, she chooses a coin to flip as follows: if she has so far flipped
a
a
a
heads out of
b
b
b
total flips, then she uses a coin with an
a
b
\frac{a}{b}
b
a
probability of landing on heads. What is the probability that after all nine flips, she has gotten six heads and three tails? p10. For any prime number
p
p
p
, let
S
p
S_p
S
p
be the sum of all the positive divisors of
3
7
p
p
37
37^pp^{37}
3
7
p
p
37
(including
1
1
1
and
3
7
p
p
37
37^pp^{37}
3
7
p
p
37
). Find the sum of all primes
p
p
p
such that
S
p
S_p
S
p
is divisible by
p
p
p
. p11. Six people are playing poker. At the beginning of the game, they have
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
, and
6
6
6
dollars, respectively. At the end of the game, nobody has lost more than a dollar, and each player has a distinct nonnegative integer dollar amount. (The total amount of money in the game remains constant.) How many distinct finishing rankings (i.e. lists of first place through sixth place) are possible? p12. Let
C
1
C_1
C
1
be a circle of radius
1
1
1
, and let
C
2
C_2
C
2
be a circle of radius
1
2
\frac12
2
1
internally tangent to
C
1
C_1
C
1
. Let
{
ω
0
,
ω
1
,
.
.
.
}
\{\omega_0, \omega_1, ... \}
{
ω
0
,
ω
1
,
...
}
be an infinite sequence of circles, such that
ω
0
\omega_0
ω
0
has radius
1
2
\frac12
2
1
and each
ω
k
\omega_k
ω
k
is internally tangent to
C
1
C_1
C
1
and externally tangent to both
C
2
C_2
C
2
and
ω
k
+
1
\omega_{k+1}
ω
k
+
1
. (The
ω
k
\omega_k
ω
k
’s are mutually distinct.) What is the radius of
ω
100
\omega_{100}
ω
100
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.