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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2013 Iran MO (2nd Round)
2013 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
Hide problems
Height intersects circumcircle and 2 equal segments
Let
M
M
M
be the midpoint of (the smaller) arc
B
C
BC
BC
in circumcircle of triangle
A
B
C
ABC
A
BC
. Suppose that the altitude drawn from
A
A
A
intersects the circle at
N
N
N
. Draw two lines through circumcenter
O
O
O
of
A
B
C
ABC
A
BC
paralell to
M
B
MB
MB
and
M
C
MC
MC
, which intersect
A
B
AB
A
B
and
A
C
AC
A
C
at
K
K
K
and
L
L
L
, respectively. Prove that
N
K
=
N
L
NK=NL
N
K
=
N
L
.
recursive sequence defined by floor function
Let
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^{\infty}
{
a
n
}
n
=
1
∞
be a sequence of positive integers for which
a
n
+
2
=
[
2
a
n
a
n
+
1
]
+
[
2
a
n
+
1
a
n
]
.
a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right].
a
n
+
2
=
[
a
n
+
1
2
a
n
]
+
[
a
n
2
a
n
+
1
]
.
Prove that there exists a positive integer
m
m
m
such that
a
m
=
4
a_m=4
a
m
=
4
and
a
m
+
1
∈
{
3
,
4
}
a_{m+1} \in\{3,4\}
a
m
+
1
∈
{
3
,
4
}
.Note.
[
x
]
[x]
[
x
]
is the greatest integer not exceeding
x
x
x
.
2
2
Hide problems
Complete set of weights
Let
n
n
n
be a natural number and suppose that
w
1
,
w
2
,
…
,
w
n
w_1, w_2, \ldots , w_n
w
1
,
w
2
,
…
,
w
n
are
n
n
n
weights . We call the set of
{
w
1
,
w
2
,
…
,
w
n
}
\{ w_1, w_2, \ldots , w_n\}
{
w
1
,
w
2
,
…
,
w
n
}
to be a Perfect Set if we can achieve all of the
1
,
2
,
…
,
W
1,2, \ldots, W
1
,
2
,
…
,
W
weights with sums of
w
1
,
w
2
,
…
,
w
n
w_1, w_2, \ldots , w_n
w
1
,
w
2
,
…
,
w
n
, where
W
=
∑
i
=
1
n
w
i
W=\sum_{i=1}^n w_i
W
=
∑
i
=
1
n
w
i
. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.
Amazing Table
Suppose a
m
×
n
m \times n
m
×
n
table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either
+
1
+1
+
1
or
−
1
-1
−
1
to all of its cells. Prove that if for all arbitrary
3
×
3
3 \times 3
3
×
3
table we can change all numbers to zero, then we can change all numbers of
m
×
n
m \times n
m
×
n
table to zero.(Hint: First of all think about it how we can change number of
3
×
3
3 \times 3
3
×
3
table to zero.)
1
2
Hide problems
a/b = a.b
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers for which
gcd
(
a
,
b
)
=
1
\gcd(a,b)=1
g
cd
(
a
,
b
)
=
1
, and
a
b
=
b
.
a
‾
\frac{a}{b}=\overline{b.a}
b
a
=
b
.
a
. (For example, if
a
=
92
a=92
a
=
92
and
b
=
13
b=13
b
=
13
, then
b
/
a
=
13.92
b/a=13.92
b
/
a
=
13.92
)
symmedian in two circles
Let
P
P
P
be a point out of circle
C
C
C
. Let
P
A
PA
P
A
and
P
B
PB
PB
be the tangents to the circle drawn from
C
C
C
. Choose a point
K
K
K
on
A
B
AB
A
B
. Suppose that the circumcircle of triangle
P
B
K
PBK
PB
K
intersects
C
C
C
again at
T
T
T
. Let
P
′
{P}'
P
′
be the reflection of
P
P
P
with respect to
A
A
A
. Prove that
∠
P
B
T
=
∠
P
′
K
A
\angle PBT = \angle {P}'KA
∠
PBT
=
∠
P
′
K
A