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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
1998 Iran MO (3rd Round)
1998 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(5)
4
1
Hide problems
Show that there exists a subset of {1, 2, ..., n}
Let be given
r
1
,
r
2
,
…
,
r
n
∈
R
r_1,r_2,\ldots,r_n \in \mathbb R
r
1
,
r
2
,
…
,
r
n
∈
R
. Show that there exists a subset
I
I
I
of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n \}
{
1
,
2
,
…
,
n
}
which which has one or two elements in common with the sets
{
i
,
i
+
1
,
i
+
2
}
,
(
1
≤
i
≤
n
−
2
)
\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)
{
i
,
i
+
1
,
i
+
2
}
,
(
1
≤
i
≤
n
−
2
)
such that \left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.
5
1
Hide problems
Prove that line AP is a median of triangle ABD
In a triangle
A
B
C
ABC
A
BC
, the bisector of angle
B
A
C
BAC
B
A
C
intersects
B
C
BC
BC
at
D
D
D
. The circle
Γ
\Gamma
Γ
through
A
A
A
which is tangent to
B
C
BC
BC
at
D
D
D
meets
A
C
AC
A
C
again at
M
M
M
. Line
B
M
BM
BM
meets
Γ
\Gamma
Γ
again at
P
P
P
. Prove that line
A
P
AP
A
P
is a median of
△
A
B
D
.
\triangle ABD.
△
A
B
D
.
3
4
Show problems
1
3
Hide problems
Every integer occurs in this sequence exactly once
Define the sequence
(
x
n
)
(x_n)
(
x
n
)
by
x
0
=
0
x_0 = 0
x
0
=
0
and for all
n
∈
N
,
n \in \mathbb N,
n
∈
N
,
x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases} where
k
∈
N
0
,
r
∈
N
k \in \mathbb N_0, r \in \mathbb N
k
∈
N
0
,
r
∈
N
. Prove that every integer occurs in this sequence exactly once.
Find all functions N -> N with three conditions
Find all functions
f
:
N
→
N
f: \mathbb N \to \mathbb N
f
:
N
→
N
such that for all positive integers
m
,
n
m,n
m
,
n
,(i)
m
f
(
f
(
m
)
)
=
(
f
(
m
)
)
2
mf(f(m))=\left( f(m) \right)^2
m
f
(
f
(
m
))
=
(
f
(
m
)
)
2
, (ii) If
gcd
(
m
,
n
)
=
d
\gcd(m,n)=d
g
cd
(
m
,
n
)
=
d
, then
f
(
m
n
)
⋅
f
(
d
)
=
d
⋅
f
(
m
)
⋅
f
(
n
)
f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)
f
(
mn
)
⋅
f
(
d
)
=
d
⋅
f
(
m
)
⋅
f
(
n
)
, (iii)
f
(
m
)
=
m
f(m)=m
f
(
m
)
=
m
if and only if
m
=
1
m=1
m
=
1
.
Table and nuts
A one-player game is played on a
m
×
n
m \times n
m
×
n
table with
m
×
n
m \times n
m
×
n
nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
for which we can remove all nuts from the table.
2
2
Hide problems
Show that the angles are equal
Let
A
B
C
D
ABCD
A
BC
D
be a convex pentagon such that
∠
D
C
B
=
∠
D
E
A
=
9
0
∘
,
and
D
C
=
D
E
.
\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.
∠
D
CB
=
∠
D
E
A
=
9
0
∘
,
and
D
C
=
D
E
.
Let
F
F
F
be a point on AB such that
A
F
:
B
F
=
A
E
:
B
C
AF:BF=AE:BC
A
F
:
BF
=
A
E
:
BC
. Show that
∠
F
E
C
=
∠
B
D
C
,
and
∠
F
C
E
=
∠
A
D
E
.
\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.
∠
FEC
=
∠
B
D
C
,
and
∠
FCE
=
∠
A
D
E
.
Convex Hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon such that
A
B
=
B
C
,
C
D
=
D
E
AB = BC, CD = DE
A
B
=
BC
,
C
D
=
D
E
and
E
F
=
F
A
EF = FA
EF
=
F
A
. Prove that
A
B
B
E
+
C
D
A
D
+
E
F
C
F
≥
3
2
.
\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.
BE
A
B
+
A
D
C
D
+
CF
EF
≥
2
3
.