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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1993 Iran MO (2nd round)
1993 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
1
1
Hide problems
Minimum number of edges [Iran Second Round 1993]
G
G
G
is a graph with
n
n
n
vertices
A
1
,
A
2
,
…
,
A
n
,
A_1,A_2,\ldots,A_n,
A
1
,
A
2
,
…
,
A
n
,
such that for each pair of non adjacent vertices
A
i
A_i
A
i
and
A
j
A_j
A
j
, there exist another vertex
A
k
A_k
A
k
that is adjacent to both
A
i
A_i
A
i
and
A
j
.
A_j .
A
j
.
(a) Find the minimum number of edges in such a graph.(b) If
n
=
6
n = 6
n
=
6
and
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A_1,A_2,A_3,A_4,A_5,
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
and
A
6
A_6
A
6
form a cycle of length
6
,
6,
6
,
find the number of edges that must be added to this cycle such that the above condition holds.
3
2
Hide problems
Find the smallest positive integer m[Iran Second Round 1993]
Let
n
,
r
n, r
n
,
r
be positive integers. Find the smallest positive integer
m
m
m
satisfying the following condition. For each partition of the set
{
1
,
2
,
…
,
m
}
\{1, 2, \ldots ,m \}
{
1
,
2
,
…
,
m
}
into
r
r
r
subsets
A
1
,
A
2
,
…
,
A
r
A_1,A_2, \ldots ,A_r
A
1
,
A
2
,
…
,
A
r
, there exist two numbers
a
a
a
and
b
b
b
in some
A
i
,
1
≤
i
≤
r
A_i, 1 \leq i \leq r
A
i
,
1
≤
i
≤
r
, such that
1
<
a
b
<
1
+
1
n
.
1 < \frac ab < 1 +\frac 1n.
1
<
b
a
<
1
+
n
1
.
The fraction f(x)/g(x) [Iran Second Round 1993]
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be two polynomials with real coefficients such that for infinitely many rational values of
x
x
x
, the fraction
f
(
x
)
g
(
x
)
\frac{f(x)}{g(x)}
g
(
x
)
f
(
x
)
is rational. Prove that
f
(
x
)
g
(
x
)
\frac{f(x)}{g(x)}
g
(
x
)
f
(
x
)
can be written as the ratio of two polynomials with rational coefficients.
2
2
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existence of a point P [Iran Second Round 1993]
Let
A
B
C
ABC
A
BC
be an acute triangle with sides and area equal to
a
,
b
,
c
a, b, c
a
,
b
,
c
and
S
S
S
respectively. [color=#FF0000]Prove or disprove that a necessary and sufficient condition for existence of a point
P
P
P
inside the triangle
A
B
C
ABC
A
BC
such that the distance between
P
P
P
and the vertices of
A
B
C
ABC
A
BC
be equal to
x
,
y
x, y
x
,
y
and
z
z
z
respectively is that there be a triangle with sides
a
,
y
,
z
a, y, z
a
,
y
,
z
and area
S
1
S_1
S
1
, a triangle with sides
b
,
z
,
x
b, z, x
b
,
z
,
x
and area
S
2
S_2
S
2
and a triangle with sides
c
,
x
,
y
c, x, y
c
,
x
,
y
and area
S
3
S_3
S
3
where
S
1
+
S
2
+
S
3
=
S
.
S_1 + S_2 + S_3 = S.
S
1
+
S
2
+
S
3
=
S
.
Infinitely many straight lines [Iran Second Round 1993]
Show that if
D
1
D_1
D
1
and
D
2
D_2
D
2
are two skew lines, then there are infinitely many straight lines such that their points have equal distance from
D
1
D_1
D
1
and
D
2
.
D_2.
D
2
.