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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1996 Iran MO (2nd round)
1996 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
4
1
Hide problems
Blue and red points lie on a line [Iran Second Round 1996]
Let
n
n
n
blue points
A
i
A_i
A
i
and
n
n
n
red points
B
i
(
i
=
1
,
2
,
…
,
n
)
B_i \ (i = 1, 2, \ldots , n)
B
i
(
i
=
1
,
2
,
…
,
n
)
be situated on a line. Prove that
∑
i
,
j
A
i
B
j
≥
∑
i
<
j
A
i
A
j
+
∑
i
<
j
B
i
B
j
.
\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.
i
,
j
∑
A
i
B
j
≥
i
<
j
∑
A
i
A
j
+
i
<
j
∑
B
i
B
j
.
1
1
Hide problems
There exists a triangle [Iran Second Round 1996]
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers. Prove that there exists a triangle with side lengths
a
,
b
,
c
a, b, c
a
,
b
,
c
if and only if
2
(
a
4
+
b
4
+
c
4
)
<
(
a
2
+
b
2
+
c
2
)
2
.
2(a^4 + b^4 + c^4) < (a^2 + b^2 + c^2)^2.
2
(
a
4
+
b
4
+
c
4
)
<
(
a
2
+
b
2
+
c
2
)
2
.
3
1
Hide problems
Mnp right triangle
Let
N
N
N
be the midpoint of side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. Right isosceles triangles
A
B
M
ABM
A
BM
and
A
C
P
ACP
A
CP
are constructed outside the triangle, with bases
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that
△
M
N
P
\triangle MNP
△
MNP
is also a right isosceles triangle.