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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2012 Iran MO (2nd Round)
2012 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
1
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Incenter inside the incircle
The incircle of triangle
A
B
C
ABC
A
BC
, is tangent to sides
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
in
D
,
E
D,E
D
,
E
and
F
F
F
respectively. The reflection of
F
F
F
with respect to
B
B
B
and the reflection of
E
E
E
with respect to
C
C
C
are
T
T
T
and
S
S
S
respectively. Prove that the incenter of triangle
A
S
T
AST
A
ST
is inside or on the incircle of triangle
A
B
C
ABC
A
BC
.Proposed by Mehdi E'tesami Fard
2
2
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Numbers around a circle
Suppose
n
n
n
is a natural number. In how many ways can we place numbers
1
,
2
,
.
.
.
.
,
n
1,2,....,n
1
,
2
,
....
,
n
around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
fourth degree polynomail, 4 variable polynomial!
Consider the second degree polynomial
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant,
a
2
−
4
b
a^2-4b
a
2
−
4
b
be greater than or equal to zero. Note that the discriminant is also a polynomial with variables
a
a
a
and
b
b
b
. Prove that the same story is not true for polynomials of degree
4
4
4
: Prove that there does not exist a
4
4
4
variable polynomial
P
(
a
,
b
,
c
,
d
)
P(a,b,c,d)
P
(
a
,
b
,
c
,
d
)
such that:The fourth degree polynomial
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
x^4+ax^3+bx^2+cx+d
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
can be written as the product of four
1
1
1
st degree polynomials if and only if
P
(
a
,
b
,
c
,
d
)
≥
0
P(a,b,c,d)\ge 0
P
(
a
,
b
,
c
,
d
)
≥
0
. (All the coefficients are real numbers.)Proposed by Sahand Seifnashri
1
2
Hide problems
Parallel and three circles
Consider a circle
C
1
C_1
C
1
and a point
O
O
O
on it. Circle
C
2
C_2
C
2
with center
O
O
O
, intersects
C
1
C_1
C
1
in two points
P
P
P
and
Q
Q
Q
.
C
3
C_3
C
3
is a circle which is externally tangent to
C
2
C_2
C
2
at
R
R
R
and internally tangent to
C
1
C_1
C
1
at
S
S
S
and suppose that
R
S
RS
RS
passes through
Q
Q
Q
. Suppose
X
X
X
and
Y
Y
Y
are second intersection points of
P
R
PR
PR
and
O
R
OR
OR
with
C
1
C_1
C
1
. Prove that
Q
X
QX
QX
is parallel with
S
Y
SY
S
Y
.
Partition of natural numbers by 2-element sets
a) Do there exist
2
2
2
-element subsets
A
1
,
A
2
,
A
3
,
.
.
.
A_1,A_2,A_3,...
A
1
,
A
2
,
A
3
,
...
of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number
n
n
n
, sum of the elements of
A
n
A_n
A
n
equals
1391
+
n
1391+n
1391
+
n
?b) Do there exist
2
2
2
-element subsets
A
1
,
A
2
,
A
3
,
.
.
.
A_1,A_2,A_3,...
A
1
,
A
2
,
A
3
,
...
of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number
n
n
n
, sum of the elements of
A
n
A_n
A
n
equals
1391
+
n
2
1391+n^2
1391
+
n
2
?Proposed by Morteza Saghafian