MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1997 Iran MO (2nd round)
1997 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
Hide problems
Swapping -1 with +1 - Iran NMO 1997 (Second Round)- Problem3
We have a
n
×
n
n\times n
n
×
n
table and we’ve written numbers
0
,
+
1
o
r
−
1
0,+1 \ or \ -1
0
,
+
1
or
−
1
in each
1
×
1
1\times1
1
×
1
square such that in every row or column, there is only one
+
1
+1
+
1
and one
−
1
-1
−
1
. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap
+
1
+1
+
1
s with
−
1
-1
−
1
s.
Maximum of p - Iran NMO 1997 (Second Round) - Problem6
Let
a
,
b
a,b
a
,
b
be positive integers and
p
=
b
4
2
a
−
b
2
a
+
b
p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}
p
=
4
b
2
a
+
b
2
a
−
b
be a prime number. Find the maximum value of
p
p
p
and justify your answer.
2
2
Hide problems
MPQ=2KML - Iran NMO 1997 (Second Round) - Problem2
Let segments
K
N
,
K
L
KN,KL
K
N
,
K
L
be tangent to circle
C
C
C
at points
N
,
L
N,L
N
,
L
, respectively.
M
M
M
is a point on the extension of the segment
K
N
KN
K
N
and
P
P
P
is the other meet point of the circle
C
C
C
and the circumcircle of
△
K
L
M
\triangle KLM
△
K
L
M
.
Q
Q
Q
is on
M
L
ML
M
L
such that
N
Q
NQ
NQ
is perpendicular to
M
L
ML
M
L
. Prove that
∠
M
P
Q
=
2
∠
K
M
L
.
\angle MPQ=2\angle KML.
∠
MPQ
=
2∠
K
M
L
.
ABC is isosceles - Iran NMO 1997 (Second Round) - Problem5
In triangle
A
B
C
ABC
A
BC
, angles
B
,
C
B,C
B
,
C
are acute. Point
D
D
D
is on the side
B
C
BC
BC
such that
A
D
⊥
B
C
AD\perp{BC}
A
D
⊥
BC
. Let the interior bisectors of
∠
B
,
∠
C
\angle B,\angle C
∠
B
,
∠
C
meet
A
D
AD
A
D
at
E
,
F
E,F
E
,
F
, respectively. If
B
E
=
C
F
BE=CF
BE
=
CF
, prove that
A
B
C
ABC
A
BC
is isosceles.
1
2
Hide problems
3x^2+x=4y^2+y - Iran NMO 1997 (Second Round) - Problem1
Let
x
,
y
x,y
x
,
y
be positive integers such that
3
x
2
+
x
=
4
y
2
+
y
3x^2+x=4y^2+y
3
x
2
+
x
=
4
y
2
+
y
. Prove that
x
−
y
x-y
x
−
y
is a perfect square.
x1x2x3x4=1 - Iran NMO 1997 (Second Round) - Problem4
Let
x
1
,
x
2
,
x
3
,
x
4
x_1,x_2,x_3,x_4
x
1
,
x
2
,
x
3
,
x
4
be positive reals such that
x
1
x
2
x
3
x
4
=
1
x_1x_2x_3x_4=1
x
1
x
2
x
3
x
4
=
1
. Prove that:
∑
i
=
1
4
x
i
3
≥
max
{
∑
i
=
1
4
x
i
,
∑
i
=
1
4
1
x
i
}
.
\sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}.
i
=
1
∑
4
x
i
3
≥
max
{
i
=
1
∑
4
x
i
,
i
=
1
∑
4
x
i
1
}
.