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Princeton University Math Competition
2018 Princeton University Math Competition
2018 PUMaC Individual Finals A
2018 PUMaC Individual Finals A
Part of
2018 Princeton University Math Competition
Subcontests
(3)
3
1
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2018 PUMaC Individual Finals A3
We say that the prime numbers
p
1
,
…
,
p
n
p_1,\dots,p_n
p
1
,
…
,
p
n
construct the graph
G
G
G
if we can assign to each vertex of
G
G
G
a natural number whose prime divisors are among
p
1
,
…
,
p
n
p_1,\dots,p_n
p
1
,
…
,
p
n
and there is an edge between two vertices in
G
G
G
if and only if the numbers assigned to the two vertices have a common divisor greater than
1
1
1
. What is the minimal
n
n
n
such that there exist prime numbers
p
1
,
…
,
p
n
p_1,\dots,p_n
p
1
,
…
,
p
n
which construct any graph
G
G
G
with
N
N
N
vertices?
2
1
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2018 PUMaC Individual Finals A2
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R^{+}}\to\mathbb{R^+}
f
:
R
+
→
R
+
such that for all
x
,
y
∈
R
+
x,y\in\mathbb{R^+}
x
,
y
∈
R
+
it holds that
f
(
x
y
(
1
x
+
1
y
+
1
x
+
y
)
)
=
f
(
x
y
(
1
x
+
1
y
)
)
+
f
(
x
)
f
(
y
x
+
y
)
.
f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).
f
(
x
y
(
x
1
+
y
1
+
x
+
y
1
)
)
=
f
(
x
y
(
x
1
+
y
1
)
)
+
f
(
x
)
f
(
x
+
y
y
)
.
1
1
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PUMaC Individual Finals A1/B3
Let
A
B
C
ABC
A
BC
be a triangle. Construct three circles
k
1
k_1
k
1
,
k
2
k_2
k
2
, and
k
3
k_3
k
3
with the same radius such that they intersect each other at a common point
O
O
O
inside the triangle
A
B
C
ABC
A
BC
and
k
1
∩
k
2
=
{
A
,
O
}
k_1\cap k_2=\{A,O\}
k
1
∩
k
2
=
{
A
,
O
}
,
k
2
∩
k
3
=
{
B
,
O
}
k_2 \cap k_3=\{B,O\}
k
2
∩
k
3
=
{
B
,
O
}
,
k
3
∩
k
1
=
{
C
,
O
}
k_3\cap k_1=\{C,O\}
k
3
∩
k
1
=
{
C
,
O
}
. Let
t
a
t_a
t
a
be a common tangent of circles
k
1
k_1
k
1
and
k
2
k_2
k
2
such that
A
A
A
is closer to
t
a
t_a
t
a
than
O
O
O
. Define
t
b
t_b
t
b
and
t
c
t_c
t
c
similarly. Those three tangents determine a triangle
M
N
P
MNP
MNP
such that the triangle
A
B
C
ABC
A
BC
is inside the triangle
M
N
P
MNP
MNP
. Prove that the area of
M
N
P
MNP
MNP
is at least
9
9
9
times the area of
A
B
C
ABC
A
BC
.