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Princeton University Math Competition
2022 Princeton University Math Competition
2022 Princeton University Math Competition
Part of
Princeton University Math Competition
Subcontests
(28)
A8
1
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2022 PUMaC Geometry A8
Let
△
A
B
C
\vartriangle ABC
△
A
BC
have sidelengths
B
C
=
7
BC = 7
BC
=
7
,
C
A
=
8
CA = 8
C
A
=
8
, and,
A
B
=
9
AB = 9
A
B
=
9
, and let
Ω
\Omega
Ω
denote the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let circles
ω
A
\omega_A
ω
A
,
ω
B
\omega_B
ω
B
,
ω
C
\omega_C
ω
C
be internally tangent to the minor arcs
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of
Ω
\Omega
Ω
, respectively, and tangent to the segments
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at points
X
X
X
,
Y
Y
Y
, and,
Z
Z
Z
, respectively. Suppose that
B
X
X
C
=
C
Y
Y
A
=
A
Z
Z
B
=
1
2
\frac{BX}{XC} = \frac{CY}{Y A} = \frac{AZ}{ZB} = \frac12
XC
BX
=
Y
A
C
Y
=
ZB
A
Z
=
2
1
. Let
t
A
B
t_{AB}
t
A
B
be the length of the common external tangent of
ω
A
\omega_A
ω
A
and
ω
B
\omega_B
ω
B
, let
t
B
C
t_{BC}
t
BC
be the length of the common external tangent of
ω
B
\omega_B
ω
B
and
ω
C
\omega_C
ω
C
, and let
t
C
A
t_{CA}
t
C
A
be the length of the common external tangent of
ω
C
\omega_C
ω
C
and
ω
A
\omega_A
ω
A
. If
t
A
B
+
t
B
C
+
t
C
A
t_{AB} + t_{BC} + t_{CA}
t
A
B
+
t
BC
+
t
C
A
can be expressed as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
, find
m
+
n
m + n
m
+
n
.
A7
1
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2022 PUMaC Geometry A7
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
B
C
=
7
BC = 7
BC
=
7
,
C
A
=
6
CA = 6
C
A
=
6
, and,
A
B
=
5
AB = 5
A
B
=
5
. Let
I
I
I
be the incenter of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let the incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
touch sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at points
D
,
E
D,E
D
,
E
, and
F
F
F
. Let the circumcircle of
△
A
E
F
\vartriangle AEF
△
A
EF
meet the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
for a second time at point
X
≠
A
X\ne A
X
=
A
. Let
P
P
P
denote the intersection of
X
I
XI
X
I
and
E
F
EF
EF
. If the product
X
P
⋅
I
P
XP \cdot IP
XP
⋅
I
P
can be written as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
, find
m
+
n
m + n
m
+
n
.
A6 / B8
1
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2022 PUMaC Geometry A6 / B8
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has sidelengths
A
B
=
10
AB = 10
A
B
=
10
,
A
C
=
14
AC = 14
A
C
=
14
, and,
B
C
=
16
BC = 16
BC
=
16
. Circle
ω
1
\omega_1
ω
1
is tangent to rays
A
B
→
\overrightarrow{AB}
A
B
,
A
C
→
\overrightarrow{AC}
A
C
and passes through
B
B
B
. Circle
ω
2
\omega_2
ω
2
is tangent to rays
A
B
→
\overrightarrow{AB}
A
B
,
A
C
→
\overrightarrow{AC}
A
C
and passes through
C
C
C
. Let
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
intersect at points
X
,
Y
X, Y
X
,
Y
. The square of the perimeter of triangle
△
A
X
Y
\vartriangle AXY
△
A
X
Y
is equal to
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
, and,
d
d
d
are positive integers such that
a
a
a
and
d
d
d
are relatively prime, and
c
c
c
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
.
A5 / B7
1
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2022 PUMaC Geometry A5 / B7
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
8
BC = 8
BC
=
8
, and,
C
A
=
7
CA = 7
C
A
=
7
. Let the center of the
A
A
A
-excircle be
O
O
O
, and let the
A
A
A
-excircle touch lines
B
C
BC
BC
,
C
A
CA
C
A
, and,
A
B
AB
A
B
at points
X
,
Y
X, Y
X
,
Y
, and,
Z
Z
Z
, respectively. Let
h
1
h_1
h
1
,
h
2
h_2
h
2
, and,
h
3
h_3
h
3
denote the distances from
O
O
O
to lines
X
Y
XY
X
Y
,
Y
Z
Y Z
Y
Z
, and, ZX, respectively. If
h
1
2
+
h
2
2
+
h
3
2
h^2_1+ h^2_2+ h^2_3
h
1
2
+
h
2
2
+
h
3
2
can be written as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
, find
m
+
n
m + n
m
+
n
.
A4 / B6
1
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2022 PUMaC Geometry A4 / B6
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle. Points
D
,
E
,
F
D,E, F
D
,
E
,
F
are drawn on sides
A
B
AB
A
B
,
B
C
BC
BC
, and
C
A
CA
C
A
respectively such that
[
A
D
F
]
=
[
B
E
D
]
+
[
C
E
F
]
[ADF] = [BED] + [CEF]
[
A
D
F
]
=
[
BE
D
]
+
[
CEF
]
and
△
A
D
F
∼
△
B
E
D
∼
△
C
E
F
\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF
△
A
D
F
∼
△
BE
D
∼
△
CEF
. The ratio
[
A
B
C
]
[
D
E
F
]
\frac{[ABC]}{[DEF]}
[
D
EF
]
[
A
BC
]
can be expressed as
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
are positive integers such that
a
a
a
and
d
d
d
are relatively prime, and
c
c
c
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. (Here
[
P
]
[P]
[
P
]
denotes the area of polygon
P
P
P
.)
A3 / B5
1
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2022 PUMaC Geometry A3 / B5
Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
. Then Dylan chooses a point
P
P
P
within the circle of radius
2
2
2
centered at the origin. Let
M
M
M
be the maximum possible value of
∣
P
A
∣
⋅
∣
P
B
∣
⋅
∣
P
C
∣
⋅
∣
P
D
∣
⋅
∣
P
E
∣
⋅
∣
P
F
∣
|PA| \cdot |PB| \cdot |PC| \cdot |PD| \cdot |PE| \cdot |PF|
∣
P
A
∣
⋅
∣
PB
∣
⋅
∣
PC
∣
⋅
∣
P
D
∣
⋅
∣
PE
∣
⋅
∣
PF
∣
, and let
N
N
N
be the number of possible points
P
P
P
for which this maximal value is obtained. Find
M
+
N
2
M + N^2
M
+
N
2
.
A2 / B4
1
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2022 PUMaC Geometry A2 / B4
An ellipse has foci
A
A
A
and
B
B
B
and has the property that there is some point
C
C
C
on the ellipse such that the area of the circle passing through
A
A
A
,
B
B
B
, and,
C
C
C
is equal to the area of the ellipse. Let
e
e
e
be the largest possible eccentricity of the ellipse. One may write
e
2
e^2
e
2
as
a
+
b
c
\frac{a+\sqrt{b}}{c}
c
a
+
b
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are integers such that
a
a
a
and
c
c
c
are relatively prime, and b is not divisible by the square of any prime. Find
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
.
A1 / B3
1
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2022 PUMaC Geometry A1 / B3
Circle
Γ
\Gamma
Γ
is centered at
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the plane with radius
2022
3
2022\sqrt3
2022
3
. Circle
Ω
\Omega
Ω
is centered on the
x
x
x
-axis, passes through the point
A
=
(
6066
,
0
)
A = (6066, 0)
A
=
(
6066
,
0
)
, and intersects
Γ
\Gamma
Γ
orthogonally at the point
P
=
(
x
,
y
)
P = (x, y)
P
=
(
x
,
y
)
with
y
>
0
y > 0
y
>
0
. If the length of the minor arc
A
P
AP
A
P
on
Ω
\Omega
Ω
can be expressed as
m
π
n
\frac{m\pi}{n}
n
mπ
forrelatively prime positive integers
m
,
n
m, n
m
,
n
, find
m
+
n
m + n
m
+
n
. (Two circles intersect orthogonally at a point
P
P
P
if the tangent lines at
P
P
P
form a right angle.)
B1
2
Hide problems
2022 PUMaC Individual Finals B1
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be real numbers for which
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2 + b^2 + c^2 + d^2 = 1
a
2
+
b
2
+
c
2
+
d
2
=
1
. Show the following inequality:
a
2
+
b
2
−
c
2
−
d
2
≤
2
+
4
(
a
c
+
b
d
)
.
a^2 + b^2 - c^2 - d^2 \le \sqrt{2 + 4(ac + bd)}.
a
2
+
b
2
−
c
2
−
d
2
≤
2
+
4
(
a
c
+
b
d
)
.
2022 PUMaC Geometry B1
A triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is situated on the plane and a point
E
E
E
is given on segment
A
C
AC
A
C
. Let
D
D
D
be a point in the plane such that lines
A
D
AD
A
D
and
B
E
BE
BE
are parallel. Suppose that
∠
E
B
C
=
2
5
o
\angle EBC = 25^o
∠
EBC
=
2
5
o
,
∠
B
C
A
=
3
2
o
\angle BCA = 32^o
∠
BC
A
=
3
2
o
, and
∠
C
A
B
=
6
0
o
\angle CAB = 60^o
∠
C
A
B
=
6
0
o
. Find the smallest possible value of
∠
D
A
B
\angle DAB
∠
D
A
B
in degrees.
A3
1
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2022 PUMaC Individual Finals A3
Let
n
n
n
be a positive integer. We call a
n
n
n
-tuple
(
a
1
,
.
.
.
,
a
n
)
(a_1, . . . , a_n)
(
a
1
,
...
,
a
n
)
of positive integers nice if
∙
\bullet
∙
g
c
d
(
a
1
,
.
.
.
,
a
n
)
=
1
gcd (a_1, . . . , a_n) = 1
g
c
d
(
a
1
,
...
,
a
n
)
=
1
, and
∙
\bullet
∙
a
i
∣
a
i
−
1
+
a
i
+
1
a_i|a_{i-1} + a_{i+1}
a
i
∣
a
i
−
1
+
a
i
+
1
, for all
i
=
1
,
.
.
.
,
n
i = 1, . . . , n
i
=
1
,
...
,
n
(we define
a
0
=
a
n
a_0 = a_n
a
0
=
a
n
and
a
n
+
1
=
a
1
a_{n+1} = a1
a
n
+
1
=
a
1
here). Find the maximal possible value of the sum
a
1
+
.
.
.
+
a
n
a_1 +...+ a_n
a
1
+
...
+
a
n
if
(
a
1
,
.
.
.
,
a
n
)
(a_1, . . . , a_n)
(
a
1
,
...
,
a
n
)
is a nice
n
n
n
-tuple.
A2 / B3
1
Hide problems
2022 PUMaC Individual Finals A2 / B3
Anna and Bob play the following game. In the beginning, Bob writes down the numbers
1
,
2
,
.
.
.
,
2022
1, 2, ... , 2022
1
,
2
,
...
,
2022
on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the
1011
1011
1011
numbers on both sides are written in some order. After Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win? For example, Bob could write down numbers in the following way:
4
,
2
,
5
,
7
,
9
,
.
.
.
,
2021
,
,
3
,
1
,
6
,
8
,
10
,
.
.
.
,
2022
4, 2, 5, 7, 9, ... , 2021\,\,\,\,\,\,\,\,\,\,,\, \,\,\,\,\,\,\,\,\,\,,\, 3, 1, 6, 8, 10, ... , 2022
4
,
2
,
5
,
7
,
9
,
...
,
2021
,
,
3
,
1
,
6
,
8
,
10
,
...
,
2022
Then Anna could swap the numbers
1
,
4
1, 4
1
,
4
and then swap
2
,
3
2, 3
2
,
3
to win. However, if Anna swapped the pairs
3
,
4
3, 4
3
,
4
and
1
,
2
1, 2
1
,
2
, the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.
A1
1
Hide problems
2022 PUMaC Individual Finals A1
Let
f
:
Z
>
0
→
Z
>
0
f : Z_{>0} \to Z_{>0}
f
:
Z
>
0
→
Z
>
0
be a function which satisfies
k
∣
f
k
(
x
)
−
x
k|f^k(x)-x
k
∣
f
k
(
x
)
−
x
for all
k
,
x
∈
Z
>
0
k, x \in Z_{>0}
k
,
x
∈
Z
>
0
and
f
(
x
)
−
x
≤
2023
f(x)-x \le 2023
f
(
x
)
−
x
≤
2023
. If
f
(
1
)
=
2000
f(1) = 2000
f
(
1
)
=
2000
, what can
f
f
f
be? Remark: Here,
f
k
(
x
)
f^k (x)
f
k
(
x
)
denotes the
k
k
k
-fold application of
f
f
f
to
x
x
x
.
B2
2
Hide problems
2022 PUMaC Individual Finals B2
Given a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,construct squares
B
A
Q
P
BAQP
B
A
QP
and
A
C
R
S
ACRS
A
CRS
outside the triangle
A
B
C
ABC
A
BC
(with vertices in that listed in counterclockwise order).Show that the line from
A
A
A
perpendicular to
B
C
BC
BC
passes through the midpoint of the segment
Q
S
QS
QS
.
2022 PUMaC Geometry B2
Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are
6
6
6
,
8
8
8
, and,
10
10
10
. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.
14
1
Hide problems
2022 PUMaC Team #14
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle. Let
Q
Q
Q
be a point in the interior of
△
A
B
C
\vartriangle ABC
△
A
BC
, and let
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
denote the feet of the altitudes from
Q
Q
Q
to sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, respectively. Suppose that
B
C
=
15
BC = 15
BC
=
15
,
∠
A
B
C
=
6
0
o
\angle ABC = 60^o
∠
A
BC
=
6
0
o
,
B
Z
=
8
BZ = 8
BZ
=
8
,
Z
Q
=
6
ZQ = 6
ZQ
=
6
, and
∠
Q
C
A
=
3
0
o
\angle QCA = 30^o
∠
QC
A
=
3
0
o
. Let line
Q
X
QX
QX
intersect the circumcircle of
△
X
Y
Z
\vartriangle XY Z
△
X
Y
Z
at the point
W
≠
X
W\ne X
W
=
X
. If the ratio
W
Y
W
Z
\frac{ WY}{WZ}
W
Z
WY
can be expressed as
p
q
\frac{p}{q}
q
p
for relatively prime positive integers
p
,
q
p, q
p
,
q
, find
p
+
q
p + q
p
+
q
.
13
1
Hide problems
2022 PUMaC Team #13
Of all functions
h
:
Z
>
0
→
Z
≥
0
h : Z_{>0} \to Z_{\ge 0}
h
:
Z
>
0
→
Z
≥
0
, choose one satisfying
h
(
a
b
)
=
a
h
(
b
)
+
b
h
(
a
)
h(ab) = ah(b) + bh(a)
h
(
ab
)
=
ah
(
b
)
+
bh
(
a
)
for all
a
,
b
∈
Z
>
0
a, b \in Z_{>0}
a
,
b
∈
Z
>
0
and
h
(
p
)
=
p
h(p) = p
h
(
p
)
=
p
for all prime numbers
p
p
p
. Find the sum of all positive integers
n
≤
100
n\le 100
n
≤
100
such that
h
(
n
)
=
4
n
h(n) = 4n
h
(
n
)
=
4
n
.
12
1
Hide problems
2022 PUMaC Team #12
Observe the set
S
=
{
(
x
,
y
)
∈
Z
2
:
∣
x
∣
≤
5
S =\{(x, y) \in Z^2 : |x| \le 5
S
=
{(
x
,
y
)
∈
Z
2
:
∣
x
∣
≤
5
and
−
10
≤
y
≤
0
}
-10 \le y\le 0\}
−
10
≤
y
≤
0
}
. Find the number of points
P
P
P
in
S
S
S
such that there exists a tangent line from
P
P
P
to the parabola
y
=
x
2
+
1
y = x^2 + 1
y
=
x
2
+
1
that can be written in the form
y
=
m
x
+
b
y = mx + b
y
=
m
x
+
b
, where
m
m
m
and
b
b
b
are integers.
11
1
Hide problems
2022 PUMaC Team #11
For the function
g
(
a
)
=
max
⏟
x
∈
R
{
cos
x
+
cos
(
x
+
π
6
)
+
cos
(
x
+
π
4
)
+
c
o
s
(
x
+
a
)
}
,
g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},
g
(
a
)
=
x
∈
R
max
{
cos
x
+
cos
(
x
+
6
π
)
+
cos
(
x
+
4
π
)
+
cos
(
x
+
a
)
}
,
let
b
∈
R
b \in R
b
∈
R
be the input that maximizes
g
g
g
. If
cos
2
b
=
m
+
n
+
p
−
q
24
\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}
cos
2
b
=
24
m
+
n
+
p
−
q
for positive integers
m
,
n
,
p
,
q
m, n, p, q
m
,
n
,
p
,
q
, find
m
+
n
+
p
+
q
m + n + p + q
m
+
n
+
p
+
q
.
15
1
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2022 PUMaC Team #15
Subsets
S
S
S
of the first 3
5
5
5
positive integers
{
1
,
2
,
3
,
.
.
.
,
35
}
\{1, 2, 3, ..., 35\}
{
1
,
2
,
3
,
...
,
35
}
are called contrived if
S
S
S
has size
4
4
4
and the sum of the squares of the elements of
S
S
S
is divisible by
7
7
7
. Find the number of contrived sets.
10
1
Hide problems
2022 PUMaC Team #10
Let
α
,
β
,
γ
∈
C
\alpha, \beta, \gamma \in C
α
,
β
,
γ
∈
C
be the roots of the polynomial
x
3
−
3
x
2
+
3
x
+
7
x^3 - 3x2 + 3x + 7
x
3
−
3
x
2
+
3
x
+
7
. For any complex number
z
z
z
, let
f
(
z
)
f(z)
f
(
z
)
be defined as follows:
f
(
z
)
=
∣
z
−
α
∣
+
∣
z
−
β
∣
+
∣
z
−
γ
∣
−
2
max
⏟
w
∈
{
α
,
β
,
γ
}
∣
z
−
w
∣
.
f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.
f
(
z
)
=
∣
z
−
α
∣
+
∣
z
−
β
∣
+
∣
z
−
γ
∣
−
2
w
∈
{
α
,
β
,
γ
max
}
∣
z
−
w
∣.
Let
A
A
A
be the area of the region bounded by the locus of all
z
∈
C
z \in C
z
∈
C
at which
f
(
z
)
f(z)
f
(
z
)
attains its global minimum. Find
⌊
A
⌋
\lfloor A \rfloor
⌊
A
⌋
.
9
1
Hide problems
2022 PUMaC Team #9
In the complex plane, let
z
1
,
z
2
,
z
3
z_1, z_2, z_3
z
1
,
z
2
,
z
3
be the roots of the polynomial
p
(
x
)
=
x
3
−
a
x
2
+
b
x
−
a
b
p(x) = x^3- ax^2 + bx - ab
p
(
x
)
=
x
3
−
a
x
2
+
b
x
−
ab
. Find the number of integers
n
n
n
between
1
1
1
and
500
500
500
inclusive that are expressible as
z
1
4
+
z
2
4
+
z
3
4
z^4_1 +z^4_2 +z^4_3
z
1
4
+
z
2
4
+
z
3
4
for some choice of positive integers
a
,
b
a, b
a
,
b
.
8
1
Hide problems
2022 PUMaC Team #8
Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers
n
n
n
in the interval
[
2
,
728
]
[2, 728]
[
2
,
728
]
such that
3
t
3^t
3
t
doesn’t divide
n
!
n!
n
!
, where
t
=
⌈
n
−
3
2
⌉
t = \left\lceil \frac{n-3}{2} \right\rceil
t
=
⌈
2
n
−
3
⌉
. Find the sum of the leftover integers in that interval modulo
1000
1000
1000
.
7
1
Hide problems
2022 PUMaC Team #7
Pick
x
,
y
,
z
x, y, z
x
,
y
,
z
to be real numbers satisfying
(
−
x
+
y
+
z
)
2
−
1
3
=
4
(
y
−
z
)
2
(-x+y+z)^2-\frac13 = 4(y-z)^2
(
−
x
+
y
+
z
)
2
−
3
1
=
4
(
y
−
z
)
2
,
(
x
−
y
+
z
)
2
−
1
4
=
4
(
z
−
x
)
2
(x-y+z)^2-\frac14 = 4(z-x)2
(
x
−
y
+
z
)
2
−
4
1
=
4
(
z
−
x
)
2
, and
(
x
+
y
−
z
)
2
−
1
5
=
4
(
x
−
y
)
2
(x+y-z)^2 -\frac15 = 4(x-y)^2
(
x
+
y
−
z
)
2
−
5
1
=
4
(
x
−
y
)
2
. If the value of
x
y
+
y
z
+
z
x
xy+yz +zx
x
y
+
yz
+
z
x
can be written as
p
q
\frac{p}{q}
q
p
for relatively prime positive integers
p
,
q
p, q
p
,
q
, find
p
+
q
p + q
p
+
q
.
6
1
Hide problems
2022 PUMaC Team #6
A sequence of integers
x
1
,
x
2
,
.
.
.
x_1, x_2, ...
x
1
,
x
2
,
...
is double-dipped if
x
n
+
2
=
a
x
n
+
1
+
b
x
n
x_{n+2} = ax_{n+1} + bx_n
x
n
+
2
=
a
x
n
+
1
+
b
x
n
for all
n
≥
1
n \ge 1
n
≥
1
and some fixed integers
a
,
b
a, b
a
,
b
. Ri begins to form a sequence by randomly picking three integers from the set
{
1
,
2
,
.
.
.
,
12
}
\{1, 2, ..., 12\}
{
1
,
2
,
...
,
12
}
, with replacement. It is known that if Ri adds a term by picking anotherelement at random from
{
1
,
2
,
.
.
.
,
12
}
\{1, 2, ..., 12\}
{
1
,
2
,
...
,
12
}
, there is at least a
1
3
\frac13
3
1
chance that his resulting four-term sequence forms the beginning of a double-dipped sequence. Given this, how many distinct three-term sequences could Ri have picked to begin with?
5
1
Hide problems
2022 PUMaC Team #5
You’re given the complex number
ω
=
e
2
i
π
/
13
+
e
10
i
π
/
13
+
e
16
i
π
/
13
+
e
24
i
π
/
13
\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}
ω
=
e
2
iπ
/13
+
e
10
iπ
/13
+
e
16
iπ
/13
+
e
24
iπ
/13
, and told it’s a root of a unique monic cubic
x
3
+
a
x
2
+
b
x
+
c
x^3 +ax^2 +bx+c
x
3
+
a
x
2
+
b
x
+
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers. Determine the value of
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
.
4
1
Hide problems
2022 PUMaC Team #4
Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with
n
n
n
planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as
P
(
1
/
n
)
f
(
n
)
+
Q
(
1
/
n
)
P(1/n)f(n) + Q(1/n)
P
(
1/
n
)
f
(
n
)
+
Q
(
1/
n
)
, where
P
,
Q
P,Q
P
,
Q
are polynomials and
f
(
n
)
=
∑
i
=
1
n
1
i
f(n) =\sum^n_{i=1}\frac{1}{i}
f
(
n
)
=
∑
i
=
1
n
i
1
. Find
P
(
2023
)
+
Q
(
2023
)
P(2023) + Q(2023)
P
(
2023
)
+
Q
(
2023
)
.
3
1
Hide problems
2022 PUMaC Team #3
Provided that
{
a
i
}
i
=
1
28
\{a_i\}^{28}_{i=1}
{
a
i
}
i
=
1
28
are the
28
28
28
distinct roots of
29
x
28
+
28
x
27
+
.
.
.
+
2
x
+
1
=
0
29x^{28} + 28x^{27} + ... + 2x + 1 = 0
29
x
28
+
28
x
27
+
...
+
2
x
+
1
=
0
, then the absolute value of
∑
i
=
1
28
1
(
1
−
a
i
)
2
\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}
∑
i
=
1
28
(
1
−
a
i
)
2
1
can be written as
p
q
\frac{p}{q}
q
p
for relatively prime positive integers
p
,
q
p, q
p
,
q
. Find
p
+
q
p + q
p
+
q
.
2
1
Hide problems
2022 PUMaC Team #2
A triangle
△
A
0
A
1
A
2
\vartriangle A_0A_1A_2
△
A
0
A
1
A
2
in the plane has sidelengths
A
0
A
1
=
7
A_0A_1 = 7
A
0
A
1
=
7
,
A
1
A
2
=
8
A_1A_2 = 8
A
1
A
2
=
8
,
A
2
A
0
=
9
A_2A_0 = 9
A
2
A
0
=
9
. For
i
≥
0
i \ge 0
i
≥
0
, given
△
A
i
A
i
+
1
A
i
+
2
\vartriangle A_iA_{i+1}A_{i+2}
△
A
i
A
i
+
1
A
i
+
2
, let
A
i
+
3
A_{i+3}
A
i
+
3
be the midpoint of
A
i
A
i
+
1
A_iA_{i+1}
A
i
A
i
+
1
and let Gi be the centroid of
△
A
i
A
i
+
1
A
i
+
2
\vartriangle A_iA_{i+1}A_{i+2}
△
A
i
A
i
+
1
A
i
+
2
. Let point
G
G
G
be the limit of the sequence of points
{
G
i
}
i
=
0
∞
\{G_i\}^{\infty}_{i=0}
{
G
i
}
i
=
0
∞
. If the distance between
G
G
G
and
G
0
G_0
G
0
can be written as
a
b
c
\frac{a\sqrt{b}}{c}
c
a
b
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers such that
a
a
a
and
c
c
c
are relatively prime and
b
b
b
is not divisible by the square of any prime, find
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
.
1
1
Hide problems
2022 PUMaC Team #1
Have
b
,
c
∈
R
b, c \in R
b
,
c
∈
R
satisfy
b
∈
(
0
,
1
)
b \in (0, 1)
b
∈
(
0
,
1
)
and
c
>
0
c > 0
c
>
0
, then let
A
,
B
A,B
A
,
B
denote the points of intersection of the line
y
=
b
x
+
c
y = bx+c
y
=
b
x
+
c
with
y
=
∣
x
∣
y = |x|
y
=
∣
x
∣
, and let
O
O
O
denote the origin of
R
2
R^2
R
2
. Let
f
(
b
,
c
)
f(b, c)
f
(
b
,
c
)
denote the area of triangle
△
O
A
B
\vartriangle OAB
△
O
A
B
. Let
k
0
=
1
2022
k_0 = \frac{1}{2022}
k
0
=
2022
1
, and for
n
≥
1
n \ge 1
n
≥
1
let
k
n
=
k
n
−
1
2
k_n = k^2_{n-1}
k
n
=
k
n
−
1
2
. If the sum
∑
n
=
1
∞
f
(
k
n
,
k
n
−
1
)
\sum^{\infty}_{n=1}f(k_n, k_{n-1})
∑
n
=
1
∞
f
(
k
n
,
k
n
−
1
)
can be written as
p
q
\frac{p}{q}
q
p
for relatively prime positive integers
p
,
q
p, q
p
,
q
, find the remainder when
p
+
q
p+q
p
+
q
is divided by 1000.