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Problems
Contests
National and Regional Contests
Iran Contests
Simurgh
2019 Simurgh
2019 Simurgh
Part of
Simurgh
Subcontests
(4)
4
1
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Polynomial
Assume that every root of polynomial
P
(
x
)
=
x
d
−
a
1
x
d
−
1
+
.
.
.
+
(
−
1
)
d
−
k
a
d
P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d
P
(
x
)
=
x
d
−
a
1
x
d
−
1
+
...
+
(
−
1
)
d
−
k
a
d
is in
[
0
,
1
]
[0,1]
[
0
,
1
]
. Show that for every
k
=
1
,
2
,
.
.
.
,
d
k = 1,2,...,d
k
=
1
,
2
,
...
,
d
the following inequality holds:
a
k
−
a
k
+
1
+
.
.
.
+
(
−
1
)
d
−
k
a
d
≥
0
a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0
a
k
−
a
k
+
1
+
...
+
(
−
1
)
d
−
k
a
d
≥
0
3
1
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Graph Problem
We call a graph symmetric, if we can put its vertices on the plane such that if the edges are segments, the graph has a reflectional symmetry with respect to a line not passing through its vertices. Find the least value of
K
K
K
such that the edges of every graph with
100
100
100
vertices, can be divided into
K
K
K
symmetric subgraphs.
2
1
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Two Tangent Circles
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB=AC
A
B
=
A
C
. Let point
Q
Q
Q
be on plane such that
A
Q
∥
B
C
AQ \parallel BC
A
Q
∥
BC
and
A
Q
=
A
B
AQ = AB
A
Q
=
A
B
. Now let the
P
P
P
be the foot of perpendicular from
Q
Q
Q
to
B
C
BC
BC
. Show that the circle with diameter
P
Q
PQ
PQ
is tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
.
1
1
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Arithmetic Table
Show that there exists a
10
×
10
10 \times 10
10
×
10
table of distinct natural numbers such that if
R
i
R_i
R
i
is equal to the multiplication of numbers of row
i
i
i
and
S
i
S_i
S
i
is equal to multiplication of numbers of column
i
i
i
, then numbers
R
1
R_1
R
1
,
R
2
R_2
R
2
, ... ,
R
10
R_{10}
R
10
make a nontrivial arithmetic sequence and numbers
S
1
S_1
S
1
,
S
2
S_2
S
2
, ... ,
S
10
S_{10}
S
10
also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to
0
0
0
).