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Princeton University Math Competition
2017 Princeton University Math Competition
2017 Princeton University Math Competition
Part of
Princeton University Math Competition
Subcontests
(29)
A2
1
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2017 PUMaC Individual Finals A2
Let
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
be a monotonically decreasing sequence of positive real numbers converging to zero. Suppose that
Σ
i
=
1
∞
a
i
i
\Sigma_{i=1}^{\infty}\frac{a_i}{i}
Σ
i
=
1
∞
i
a
i
diverges. Show that
Σ
i
=
1
∞
a
i
2
2017
\Sigma_{i=1}^{\infty}a_i^{2^{2017}}
Σ
i
=
1
∞
a
i
2
2017
also diverges. You may assume in your proof that
Σ
i
=
1
∞
1
i
p
\Sigma_{i=1}^{\infty}\frac{1}{i^p}
Σ
i
=
1
∞
i
p
1
converges for all real numbers
p
>
1
p > 1
p
>
1
. (A sum
Σ
i
=
1
∞
b
i
\Sigma_{i=1}^{\infty}b_i
Σ
i
=
1
∞
b
i
of positive real numbers
b
i
b_i
b
i
diverges if for each real number
N
N
N
there is a positive integer
k
k
k
such that
b
1
+
b
2
+
.
.
.
+
b
k
>
N
b_1+b_2+...+b_k > N
b
1
+
b
2
+
...
+
b
k
>
N
.)
9
1
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2017 PUMaC Team 9
The set
{
(
x
,
y
)
∈
R
2
∣
⌊
x
+
y
⌋
⋅
⌈
x
+
y
⌉
=
(
⌊
x
⌋
+
⌈
y
⌉
)
(
⌈
x
⌉
+
⌊
y
⌋
)
,
0
≤
x
,
y
≤
100
}
\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil + \lfloor y\rfloor), 0 \le x, y \le 100\}
{(
x
,
y
)
∈
R
2
∣
⌊
x
+
y
⌋
⋅
⌈
x
+
y
⌉
=
(⌊
x
⌋
+
⌈
y
⌉)
(⌈
x
⌉
+
⌊
y
⌋)
,
0
≤
x
,
y
≤
100
}
can be thought of as a collection of line segments in the plane. If the total length of those line segments is
a
+
b
c
a + b\sqrt{c}
a
+
b
c
for
c
c
c
squarefree, find
a
+
b
+
c
a + b + c
a
+
b
+
c
. (
⌊
z
⌋
\lfloor z\rfloor
⌊
z
⌋
is the greatest integer less than or equal to
z
z
z
, and
⌈
z
⌉
\lceil z \rceil
⌈
z
⌉
is the least integer greater than or equal to
z
z
z
, for
z
∈
R
z \in R
z
∈
R
.)
16
1
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2017 PUMaC Team 16
Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers
1
1
1
gallon/unit
2
^2
2
, and Robert starts at
(
1
,
0
)
(1, 0)
(
1
,
0
)
. Each second, he moves in a straight line from the point
(
cos
(
θ
)
,
sin
(
θ
)
)
(\cos(\theta),\sin(\theta))
(
cos
(
θ
)
,
sin
(
θ
))
to the point
(
cos
(
θ
+
a
)
,
sin
(
θ
+
a
)
)
(\cos(\theta+a),\sin(\theta+a))
(
cos
(
θ
+
a
)
,
sin
(
θ
+
a
))
, where a changes after each movement. a starts out as
25
3
o
253^o
25
3
o
and decreases by
2
o
2^o
2
o
each step. If he takes
89
89
89
steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as
a
−
b
c
cot
1
o
\frac{\sqrt{a}- \sqrt{b}}{c} \cot 1^o
c
a
−
b
cot
1
o
, where
a
,
b
a, b
a
,
b
and
c
c
c
are positive integers and no prime divisor of
c
c
c
divides both
a
a
a
and
b
b
b
twice. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
.
17
1
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2017 PUMaC Team 17
Zack keeps cutting the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than
3
5
\frac35
5
3
. The expected number of cuts that Zack makes can be written as
p
q
\frac{p}{q}
q
p
for
p
p
p
and
q
q
q
relatively prime positive integers. Find
p
+
q
p + q
p
+
q
.
15
1
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2017 PUMaC Team 15
How many ordered pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfy
y
x
y
=
y
2017
yx^y = y^{2017}
y
x
y
=
y
2017
?
14
1
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2017 PUMaC Team 14
Eric rolls a ten-sided die (with sides labeled
1
1
1
through
10
10
10
) repeatedly until it lands on
3
,
5
3, 5
3
,
5
, or
7
7
7
. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Compute
m
+
n
m + n
m
+
n
.
12
1
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2017 PUMaC Team 12
Call a positive integer
n
n
n
tubular if for any two distinct primes
p
p
p
and
q
q
q
dividing
n
,
(
p
+
q
)
∣
n
n, (p + q) | n
n
,
(
p
+
q
)
∣
n
. Find the number of tubular numbers less than
100
,
000
100,000
100
,
000
. (Integer powers of primes, including
1
,
3
1, 3
1
,
3
, and
16
16
16
, are not considered tubular.)
11
1
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2017 PUMaC Team 11
For a sequence of
10
10
10
coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the Tiger number of the sequence of flips. How many such sequences have a Tiger number of
24
24
24
?
10
1
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2017 PUMaC Team 10
Given a positive integer
x
≤
233
x \le 233
x
≤
233
, let
a
a
a
be the remainder when
x
1943
x^{1943}
x
1943
is divided by
233
233
233
. Find the sum of all possible values of
a
a
a
.
8
1
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2017 PUMaC Team 8
Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with
17
17
17
orange and
5
5
5
black marshmallows. If
p
q
\frac{p}{q}
q
p
is the expected number of marshmallows remaining the instant that there is only one color left, and
p
p
p
and
q
q
q
are relatively prime positive integers, find
p
+
q
p + q
p
+
q
.
7
1
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2017 PUMaC Team 7
2017
2017
2017
voters vote by submitting a ranking of the integers
{
1
,
2
,
.
.
.
,
38
}
\{1, 2, ..., 38\}
{
1
,
2
,
...
,
38
}
from favorite (a vote for that value in
1
1
1
st place) to least favorite (a vote for that value in
38
38
38
th/last place). Let
a
k
a_k
a
k
be the integer that received the most
k
k
k
th place votes (the smallest such integer if there is a tie). Find the maximum possible value of
Σ
k
=
1
38
a
k
\Sigma_{k=1}^{38} a_k
Σ
k
=
1
38
a
k
.
5
1
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2017 PUMaC Team 5
Define the sequences
a
n
a_n
a
n
and
b
n
b_n
b
n
as follows:
a
1
=
2017
a_1 = 2017
a
1
=
2017
and
b
1
=
1
b_1 = 1
b
1
=
1
. For
n
>
1
n > 1
n
>
1
, if there is a greatest integer
k
>
1
k > 1
k
>
1
such that
a
n
a_n
a
n
is a perfect
k
k
k
th power, then
a
n
+
1
=
a
n
k
a_{n+1} =\sqrt[k]{a_n}
a
n
+
1
=
k
a
n
, otherwise
a
n
+
1
=
a
n
+
b
n
a_{n+1} = a_n + b_n
a
n
+
1
=
a
n
+
b
n
. If
a
n
+
1
≥
a
n
a_{n+1} \ge a_n
a
n
+
1
≥
a
n
then
b
n
+
1
=
b
n
b_{n+1} = b_n
b
n
+
1
=
b
n
, otherwise
b
n
+
1
=
b
n
+
1
b_{n+1} = b_n + 1
b
n
+
1
=
b
n
+
1
. Find
a
2017
a_{2017}
a
2017
.
4
1
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2017 PUMaC Team 4
Ayase chooses three numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
independently and uniformly from the interval
[
−
1
,
1
]
[-1, 1]
[
−
1
,
1
]
. The probability that
0
<
a
+
b
<
a
<
a
+
b
+
c
0 < a + b < a < a + b + c
0
<
a
+
b
<
a
<
a
+
b
+
c
can be expressed in the form
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. What is
p
+
q
p + q
p
+
q
?
3
1
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2017 PUMaC Team 3
Let
f
(
x
)
=
(
x
−
5
)
(
x
−
12
)
f(x) = (x - 5)(x - 12)
f
(
x
)
=
(
x
−
5
)
(
x
−
12
)
and
g
(
x
)
=
(
x
−
6
)
(
x
−
10
)
g(x) = (x - 6)(x - 10)
g
(
x
)
=
(
x
−
6
)
(
x
−
10
)
. Find the sum of all integers
n
n
n
such that
f
(
g
(
n
)
)
f
(
n
)
2
\frac{f(g(n))}{f(n)^2}
f
(
n
)
2
f
(
g
(
n
))
is defined and an integer.
2
1
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2017 PUMaC Team 2
Let
a
%
b
a\%b
a
%
b
denote the remainder when
a
a
a
is divided by
b
b
b
. Find
Σ
i
=
1
100
(
100
%
i
)
\Sigma_{i=1}^{100}(100\%i)
Σ
i
=
1
100
(
100%
i
)
.
1
1
Hide problems
2017 PUMaC Team 1
Call an ordered triple
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of integers feral if
b
−
a
,
c
−
a
b -a, c - a
b
−
a
,
c
−
a
and
c
−
b
c - b
c
−
b
are all prime. Find the number of feral triples where
1
≤
a
<
b
<
c
≤
20
1 \le a < b < c \le 20
1
≤
a
<
b
<
c
≤
20
.
6
1
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2017 PUMaC Team 6
In regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, let
O
∈
C
E
O \in CE
O
∈
CE
be the center of circle
Γ
\Gamma
Γ
tangent to
D
A
DA
D
A
and
D
E
DE
D
E
.
Γ
\Gamma
Γ
meets
D
E
DE
D
E
at
X
X
X
and
D
A
DA
D
A
at
Y
Y
Y
. Let the altitude from
B
B
B
meet
C
D
CD
C
D
at
P
P
P
. If
C
P
=
1
CP = 1
CP
=
1
, the area of
△
C
O
Y
\vartriangle COY
△
CO
Y
can be written in the form
a
b
sin
c
o
cos
2
c
o
\frac{a}{b} \frac{\sin c^o}{\cos^2 c^o}
b
a
c
o
s
2
c
o
s
i
n
c
o
, where
a
a
a
and
b
b
b
are relatively prime positive integers and
c
c
c
is an integer in the range
(
0
,
90
)
(0, 90)
(
0
,
90
)
. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
.
13
1
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2017 PUMaC Team 13
A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length
1
1
1
. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.
A3
1
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2017 PUMaC Individual Finals A3
Triangle
A
B
C
ABC
A
BC
has incenter
I
I
I
. The line through
I
I
I
perpendicular to
A
I
AI
A
I
meets the circumcircle of
A
B
C
ABC
A
BC
at points
P
P
P
and
Q
Q
Q
, where
P
P
P
and
B
B
B
are on the same side of
A
I
AI
A
I
. Let
X
X
X
be the point such that
P
X
PX
PX
//
C
I
CI
C
I
and
Q
X
QX
QX
//
B
I
BI
B
I
. Show that
P
B
,
Q
C
P B, QC
PB
,
QC
, and
I
X
IX
I
X
intersect at a common point.
A8
4
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A7
4
Show problems
A6/B8
4
Show problems
A5/B7
4
Show problems
A4/B6
4
Show problems
A3/B5
4
Show problems
A2/B4
4
Show problems
A1/B3
5
Show problems
B2
5
Show problems
B1
5
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