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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1995 Iran MO (2nd round)
1995 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
Hide problems
Find number of participants [Iran Second Round 1995]
Let
k
k
k
be a positive integer.
12
k
12k
12
k
persons have participated in a party and everyone shake hands with
3
k
+
6
3k+6
3
k
+
6
other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find
k
.
k.
k
.
Planes pass through a common point [ Iran Second Round 1995]
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
let
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
and
D
′
D'
D
′
be the circumcenters of the triangles
B
C
D
,
C
D
A
,
D
A
B
BCD, CDA, DAB
BC
D
,
C
D
A
,
D
A
B
and
A
B
C
ABC
A
BC
, respectively. Denote by
S
(
X
,
Y
Z
)
S(X, YZ)
S
(
X
,
Y
Z
)
the plane which passes through the point
X
X
X
and is perpendicular to the line
Y
Z
.
YZ.
Y
Z
.
Prove that if
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
and
D
′
D'
D
′
don't lie in a plane, then four planes
S
(
A
,
C
′
D
′
)
,
S
(
B
,
A
′
D
′
)
,
S
(
C
,
A
′
B
′
)
S(A, C'D'), S(B, A'D'), S(C, A'B')
S
(
A
,
C
′
D
′
)
,
S
(
B
,
A
′
D
′
)
,
S
(
C
,
A
′
B
′
)
and
S
(
D
,
B
′
C
′
)
S(D, B'C')
S
(
D
,
B
′
C
′
)
pass through a common point.
2
2
Hide problems
The incenter lies on circumcircle [Iran Second Round 95]
Let
A
B
C
ABC
A
BC
be an acute triangle and let
ℓ
\ell
ℓ
be a line in the plane of triangle
A
B
C
.
ABC.
A
BC
.
We've drawn the reflection of the line
ℓ
\ell
ℓ
over the sides
A
B
,
B
C
AB, BC
A
B
,
BC
and
A
C
AC
A
C
and they intersect in the points
A
′
,
B
′
A', B'
A
′
,
B
′
and
C
′
.
C'.
C
′
.
Prove that the incenter of the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
lies on the circumcircle of the triangle
A
B
C
.
ABC.
A
BC
.
Ceil function [Iran Second Round 1995]
Let
n
≥
0
n \geq 0
n
≥
0
be an integer. Prove that
⌈
n
+
n
+
1
+
n
+
2
⌉
=
⌈
9
n
+
8
⌉
\lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil \sqrt{9n+8} \rceil
⌈
n
+
n
+
1
+
n
+
2
⌉
=
⌈
9
n
+
8
⌉
Where
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
is the smallest integer which is greater or equal to
x
.
x.
x
.
1
1
Hide problems
Prove that there exist two sets A, B [Iran Second Round 95]
Prove that for every positive integer
n
≥
3
n \geq 3
n
≥
3
there exist two sets
A
=
{
x
1
,
x
2
,
…
,
x
n
}
A =\{ x_1, x_2,\ldots, x_n\}
A
=
{
x
1
,
x
2
,
…
,
x
n
}
and
B
=
{
y
1
,
y
2
,
…
,
y
n
}
B =\{ y_1, y_2,\ldots, y_n\}
B
=
{
y
1
,
y
2
,
…
,
y
n
}
for whichi)
A
∩
B
=
∅
.
A \cap B = \varnothing.
A
∩
B
=
∅
.
ii)
x
1
+
x
2
+
⋯
+
x
n
=
y
1
+
y
2
+
⋯
+
y
n
.
x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.
x
1
+
x
2
+
⋯
+
x
n
=
y
1
+
y
2
+
⋯
+
y
n
.
ii)
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
y
1
2
+
y
2
2
+
⋯
+
y
n
2
.
x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
y
1
2
+
y
2
2
+
⋯
+
y
n
2
.