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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2021 Iran MO (2nd Round)
2021 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(6)
6
1
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Cyclic permutation of 1400 numbers having certain property
Is it possible to arrange 1400 positive integer ( not necessarily distinct ) ,at least one of them being 2021 , around a circle such that any number on this circle equals to the sum of gcd of the two previous numbers and two next numbers? for example , if
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
are five consecutive numbers on this circle ,
c
=
gcd
(
a
,
b
)
+
gcd
(
d
,
e
)
c=\gcd(a,b)+\gcd(d,e)
c
=
g
cd
(
a
,
b
)
+
g
cd
(
d
,
e
)
5
1
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Three numbers among 1400 given numbers have certain property
1400 real numbers are given. Prove that one can choose three of them like
x
,
y
,
z
x,y,z
x
,
y
,
z
such that :
∣
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
x
4
+
y
4
+
z
4
+
1
∣
<
0.009
\left|\frac{(x-y)(y-z)(z-x)}{x^4+y^4+z^4+1}\right| < 0.009
x
4
+
y
4
+
z
4
+
1
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
<
0.009
4
1
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points contained in two different circles
n
n
n
points are given on a circle
ω
\omega
ω
. There is a circle with radius smaller than
ω
\omega
ω
such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of
ω
\omega
ω
with endpoints not belonging to the given points such that all the
n
n
n
given points remain in one side of the diameter.
3
1
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tangency implying cyclic quadrilateral
Circle
ω
\omega
ω
is inscribed in quadrilateral
A
B
C
D
ABCD
A
BC
D
and is tangent to segments
B
C
,
A
D
BC, AD
BC
,
A
D
at
E
,
F
E,F
E
,
F
, respectively.
D
E
DE
D
E
intersects
ω
\omega
ω
for the second time at
X
X
X
. if the circumcircle of triangle
D
F
X
DFX
D
FX
is tangent to lines
A
B
AB
A
B
and
C
D
CD
C
D
, prove that quadrilateral
A
F
X
C
AFXC
A
FXC
is cyclic.
2
1
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Fantastic numbers
Call a positive integer
n
n
n
"Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of
n
n
n
. ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
1
1
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Two Points on a Line , can we reach the midpoint?
There are two distinct Points
A
A
A
and
B
B
B
on a line. We color a point
P
P
P
on segment
A
B
AB
A
B
, distinct from
A
,
B
A,B
A
,
B
and midpoint of segment
A
B
AB
A
B
to red. In each move , we can reflect one of the red point wrt
A
A
A
or
B
B
B
and color the midpoint of the resulting point and the point we reflected from ( which is one of
A
A
A
or
B
B
B
) to red. For example , if we choose
P
P
P
and the reflection of
P
P
P
wrt to
A
A
A
is
P
′
P'
P
′
, then midpoint of
A
P
′
AP'
A
P
′
would be red. Is it possible to make the midpoint of
A
B
AB
A
B
red after a finite number of moves?