MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2011 Iran MO (2nd Round)
2011 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
1
2
Hide problems
points with distance at least 10 meters
We have a line and
1390
1390
1390
points around it such that the distance of each point to the line is less than
1
1
1
centimeters and the distance between any two points is more than
2
2
2
centimeters. prove that there are two points such that their distance is at least
10
10
10
meters (
1000
1000
1000
centimeters).
smallest integer n with n numbers in (-1,1)
find the smallest natural number
n
n
n
such that there exists
n
n
n
real numbers in the interval
(
−
1
,
1
)
(-1,1)
(
−
1
,
1
)
such that their sum equals zero and the sum of their squares equals
20
20
20
.
2
2
Hide problems
angle less than 30 degree
In triangle
A
B
C
ABC
A
BC
, we have
∠
A
B
C
=
60
\angle ABC=60
∠
A
BC
=
60
. The line through
B
B
B
perpendicular to side
A
B
AB
A
B
intersects angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
in
D
D
D
and the line through
C
C
C
perpendicular
B
C
BC
BC
intersects angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
in
E
E
E
. prove that
∠
B
E
D
≤
30
\angle BED\le 30
∠
BE
D
≤
30
.
maximum days rainbow can live
rainbow is the name of a bird. this bird has
n
n
n
colors and it's colors in two consecutive days are not equal. there doesn't exist
4
4
4
days in this bird's life like
i
,
j
,
k
,
l
i,j,k,l
i
,
j
,
k
,
l
such that
i
<
j
<
k
<
l
i<j<k<l
i
<
j
<
k
<
l
and the bird has the same color in days
i
i
i
and
k
k
k
and the same color in days
j
j
j
and
l
l
l
different from the colors it has in days
i
i
i
and
k
k
k
. what is the maximum number of days rainbow can live in terms of
n
n
n
?
3
2
Hide problems
BD'+CE'=D'E' if BD+CE=DE
The line
l
l
l
intersects the extension of
A
B
AB
A
B
in
D
D
D
(
D
D
D
is nearer to
B
B
B
than
A
A
A
) and the extension of
A
C
AC
A
C
in
E
E
E
(
E
E
E
is nearer to
C
C
C
than
A
A
A
) of triangle
A
B
C
ABC
A
BC
. Suppose that reflection of line
l
l
l
to perpendicular bisector of side
B
C
BC
BC
intersects the mentioned extensions in
D
′
D'
D
′
and
E
′
E'
E
′
respectively. Prove that if
B
D
+
C
E
=
D
E
BD+CE=DE
B
D
+
CE
=
D
E
, then
B
D
′
+
C
E
′
=
D
′
E
′
BD'+CE'=D'E'
B
D
′
+
C
E
′
=
D
′
E
′
.
increasing sequence
Find all increasing sequences
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
of natural numbers such that for each
i
,
j
∈
N
i,j\in \mathbb N
i
,
j
∈
N
, number of the divisors of
i
+
j
i+j
i
+
j
and
a
i
+
a
j
a_i+a_j
a
i
+
a
j
is equal. (an increasing sequence is a sequence that if
i
≤
j
i\le j
i
≤
j
, then
a
i
≤
a
j
a_i\le a_j
a
i
≤
a
j
.)