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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2020 Iran MO (2nd Round)
2020 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(6)
P6
1
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Circle divided into 2n equal sections
Divide a circle into
2
n
2n
2
n
equal sections. We call a circle filled if it is filled with the numbers
0
,
1
,
2
,
…
,
n
−
1
0,1,2,\dots,n-1
0
,
1
,
2
,
…
,
n
−
1
. We call a filled circle good if it has the following properties:
i
i
i
. Each number
0
≤
a
≤
n
−
1
0 \leq a \leq n-1
0
≤
a
≤
n
−
1
is used exactly twice
i
i
ii
ii
. For any
a
a
a
we have that there are exactly
a
a
a
sections between the two sections that have the number
a
a
a
in them. Here is an example of a good filling for
n
=
5
n=5
n
=
5
(View attachment) Prove that there doesn’t exist a good filling for
n
=
1399
n=1399
n
=
1399
P5
1
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Dividing a set into pairs
Call a pair of integers
a
a
a
and
b
b
b
square makers , if
a
b
+
1
ab+1
ab
+
1
is a perfect square. Determine for which
n
n
n
is it possible to divide the set
{
1
,
2
,
…
,
2
n
}
\{1,2, \dots , 2n\}
{
1
,
2
,
…
,
2
n
}
into
n
n
n
pairs of square makers.
P4
1
Hide problems
Prove CAD<90
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be two circles that intersect at point
A
A
A
and
B
B
B
. Define point
X
X
X
on
ω
1
\omega_1
ω
1
and point
Y
Y
Y
on
ω
2
\omega_2
ω
2
such that the line
X
Y
XY
X
Y
is tangent to both circles and is closer to
B
B
B
. Define points
C
C
C
and
D
D
D
the reflection of
B
B
B
WRT
X
X
X
and
Y
Y
Y
respectively. Prove that the angle
∠
C
A
D
\angle{CAD}
∠
C
A
D
is less than
9
0
∘
90^{\circ}
9
0
∘
P1
1
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Find the min number of partition
Let
S
S
S
is a finite set with
n
n
n
elements. We divided
A
S
AS
A
S
to
m
m
m
disjoint parts such that if
A
A
A
,
B
B
B
,
A
∪
B
A \cup B
A
∪
B
are in the same part, then
A
=
B
.
A=B.
A
=
B
.
Find the minimum value of
m
m
m
.
P3
1
Hide problems
beautiful geometry problem.
let
ω
1
\omega_1
ω
1
be a circle with
O
1
O_1
O
1
as its center , let
ω
2
\omega_2
ω
2
be a circle passing through
O
1
O_1
O
1
with center
O
2
O_2
O
2
let
A
A
A
be one of the intersection of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
let
x
x
x
be a line tangent line to
ω
1
\omega_1
ω
1
passing from
A
A
A
let
ω
3
\omega_3
ω
3
be a circle passing through
O
1
,
O
2
O_1,O_2
O
1
,
O
2
with its center on the line
x
x
x
and intersect
ω
2
\omega_2
ω
2
at
P
P
P
(not
O
1
O_1
O
1
) prove that the reflection of
P
P
P
through
x
x
x
is on
ω
1
\omega_1
ω
1
P2
1
Hide problems
Mix the frog out of it
let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals , such that
x
+
y
+
z
=
1399
x+y+z=1399
x
+
y
+
z
=
1399
find the
max
(
[
x
]
y
+
[
y
]
z
+
[
z
]
x
)
\max( [x]y + [y]z + [z]x )
max
([
x
]
y
+
[
y
]
z
+
[
z
]
x
)
(
[
a
]
[a]
[
a
]
is the biggest integer not exceeding
a
a
a
)