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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2005 Iran MO (3rd Round)
2005 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(6)
6
1
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Banach Theorem
Suppose
A
⊆
R
m
A\subseteq \mathbb R^m
A
⊆
R
m
is closed and non-empty. Let
f
:
A
→
A
f:A\to A
f
:
A
→
A
is a lipchitz function with constant less than 1. (ie there exist
c
<
1
c<1
c
<
1
that
∣
f
(
x
)
−
f
(
y
)
∣
<
c
∣
x
−
y
∣
,
∀
x
,
y
∈
A
)
|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)
∣
f
(
x
)
−
f
(
y
)
∣
<
c
∣
x
−
y
∣
,
∀
x
,
y
∈
A
)
. Prove that there exists a unique point
x
∈
A
x\in A
x
∈
A
such that
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
.
5
3
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a,b,c
Suppose
a
,
b
,
c
∈
R
+
a,b,c \in \mathbb R^+
a
,
b
,
c
∈
R
+
and
1
a
2
+
1
+
1
b
2
+
1
+
1
c
2
+
1
=
2
\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2
a
2
+
1
1
+
b
2
+
1
1
+
c
2
+
1
1
=
2
Prove that
a
b
+
a
c
+
b
c
≤
3
2
ab+ac+bc\leq \frac32
ab
+
a
c
+
b
c
≤
2
3
Concurrent
Suppose
H
H
H
and
O
O
O
are orthocenter and circumcenter of triangle
A
B
C
ABC
A
BC
.
ω
\omega
ω
is circumcircle of
A
B
C
ABC
A
BC
.
A
O
AO
A
O
intersects with
ω
\omega
ω
at
A
1
A_1
A
1
.
A
1
H
A_1H
A
1
H
intersects with
ω
\omega
ω
at
A
′
A'
A
′
and
A
′
′
A''
A
′′
is the intersection point of
ω
\omega
ω
and
A
H
AH
A
H
. We define points
B
′
,
B
′
′
,
C
′
B',\ B'',\ C'
B
′
,
B
′′
,
C
′
and
C
′
′
C''
C
′′
similiarly. Prove that
A
′
A
′
′
,
B
′
B
′
′
A'A'',B'B''
A
′
A
′′
,
B
′
B
′′
and
C
′
C
′
′
C'C''
C
′
C
′′
are concurrent in a point on the Euler line of triangle
A
B
C
ABC
A
BC
.
a^+b^n-c^n
Let
a
,
b
,
c
∈
N
a,b,c\in \mathbb N
a
,
b
,
c
∈
N
be such that
a
,
b
≠
c
a,b\neq c
a
,
b
=
c
. Prove that there are infinitely many prime numbers
p
p
p
for which there exists
n
∈
N
n\in\mathbb N
n
∈
N
that
p
∣
a
n
+
b
n
−
c
n
p|a^n+b^n-c^n
p
∣
a
n
+
b
n
−
c
n
.
4
5
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3
5
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2
5
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1
5
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