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National and Regional Contests
Iran Contests
Iran Team Selection Test
2016 Iran Team Selection Test
2016 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(5)
6
1
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You gotta have a math observation even in a council
Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties:
∙
\bullet
∙
Ascension:If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable.
∙
\bullet
∙
Symmetry: If all of the members change their vote, the final decision will change too.Prove that the council uses a weighted decision-making method ; that is , nonnegative weights
ω
1
,
ω
2
,
⋯
,
ω
5
\omega _1 , \omega _2 , \cdots ,\omega _5
ω
1
,
ω
2
,
⋯
,
ω
5
can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest.Remark. The statement isn't true at all if you replace
5
5
5
with arbitrary
n
n
n
. In fact , finding a counter example for
n
=
6
n=6
n
=
6
, was appeared in the same year's [url=https://artofproblemsolving.com/community/c6h1459567p8417532]Iran MO 2nd round P6
1
1
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functions in a table
A real function has been assigned to every cell of an
n
×
n
n \times n
n
×
n
table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.
5
2
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distances from the sides
Let
P
P
P
and
P
′
P '
P
′
be two unequal regular
n
−
n-
n
−
gons and
A
A
A
and
A
′
A'
A
′
two points inside
P
P
P
and
P
′
P '
P
′
, respectively.Suppose
{
d
1
,
d
2
,
⋯
d
n
}
\{ d_1 , d_2 , \cdots d_n \}
{
d
1
,
d
2
,
⋯
d
n
}
are the distances from
A
A
A
to the vertices of
P
P
P
and
{
d
1
′
,
d
2
′
,
⋯
d
n
′
}
\{ d'_1 , d'_2 , \cdots d'_n \}
{
d
1
′
,
d
2
′
,
⋯
d
n
′
}
are defines similarly for
P
′
,
A
′
P',A'
P
′
,
A
′
. Is it possible for
{
d
1
′
,
d
2
′
,
⋯
d
n
′
}
\{ d'_1 , d'_2 , \cdots d'_n \}
{
d
1
′
,
d
2
′
,
⋯
d
n
′
}
to be a permutation of
{
d
1
,
d
2
,
⋯
d
n
}
\{ d_1 , d_2 , \cdots d_n \}
{
d
1
,
d
2
,
⋯
d
n
}
?
Iran Team Selection Test 2016
Let
A
D
,
B
F
,
C
E
AD,BF,CE
A
D
,
BF
,
CE
be altitudes of triangle
A
B
C
ABC
A
BC
.
Q
Q
Q
is a point on
E
F
EF
EF
such that
Q
F
=
D
E
QF=DE
QF
=
D
E
and
F
F
F
is between
E
,
Q
E,Q
E
,
Q
.
P
P
P
is a point on
E
F
EF
EF
such that
E
P
=
D
F
EP=DF
EP
=
D
F
and
E
E
E
is between
P
,
F
P,F
P
,
F
.Perpendicular bisector of
D
Q
DQ
D
Q
intersect with
A
B
AB
A
B
at
X
X
X
and perpendicular bisector of
D
P
DP
D
P
intersect with
A
C
AC
A
C
at
Y
Y
Y
.Prove that midpoint of
B
C
BC
BC
lies on
X
Y
XY
X
Y
.
3
1
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Iran Team Selection Test 2016
Let
p
≠
13
p \neq 13
p
=
13
be a prime number of the form
8
k
+
5
8k+5
8
k
+
5
such that
39
39
39
is a quadratic non-residue modulo
p
p
p
. Prove that the equation
x
1
4
+
x
2
4
+
x
3
4
+
x
4
4
≡
0
(
m
o
d
p
)
x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p
x
1
4
+
x
2
4
+
x
3
4
+
x
4
4
≡
0
(
mod
p
)
has a solution in integers such that
p
∤
x
1
x
2
x
3
x
4
p\nmid x_1x_2x_3x_4
p
∤
x
1
x
2
x
3
x
4
.
2
2
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Iran Team Selection Test 2016
Let
A
B
C
ABC
A
BC
be an arbitrary triangle and
O
O
O
is the circumcenter of
△
A
B
C
\triangle {ABC}
△
A
BC
.Points
X
,
Y
X,Y
X
,
Y
lie on
A
B
,
A
C
AB,AC
A
B
,
A
C
,respectively such that the reflection of
B
C
BC
BC
WRT
X
Y
XY
X
Y
is tangent to circumcircle of
△
A
X
Y
\triangle {AXY}
△
A
X
Y
.Prove that the circumcircle of triangle
A
X
Y
AXY
A
X
Y
is tangent to circumcircle of triangle
B
O
C
BOC
BOC
.
Iran Team Selection Test 2016
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers such that
1
a
+
1
+
1
b
+
1
+
1
c
+
1
+
1
d
+
1
=
2
\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2
a
+
1
1
+
b
+
1
1
+
c
+
1
1
+
d
+
1
1
=
2
. Prove that
∑
c
y
c
a
2
+
1
2
≥
(
3.
∑
c
y
c
a
)
−
8
\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8
cyc
∑
2
a
2
+
1
≥
(
3.
cyc
∑
a
)
−
8