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National and Regional Contests
Indonesia Contests
Indonesia TST
2008 Indonesia TST
2008 Indonesia TST
Part of
Indonesia TST
Subcontests
(4)
2
4
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P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}, at least 1 real root
A polynomial
P
(
x
)
=
1
+
x
2
+
x
5
+
x
n
1
+
.
.
.
+
x
n
s
+
x
2008
P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}
P
(
x
)
=
1
+
x
2
+
x
5
+
x
n
1
+
...
+
x
n
s
+
x
2008
with
n
1
,
.
.
.
,
n
s
n_1, ..., n_s
n
1
,
...
,
n
s
are positive integers and
5
<
n
1
<
.
.
.
<
n
s
<
2008
5 < n_1 < ... <n_s < 2008
5
<
n
1
<
...
<
n
s
<
2008
are given. Prove that if
P
(
x
)
P(x)
P
(
x
)
has at least a real root, then the root is not greater than
1
−
5
2
\frac{1-\sqrt5}{2}
2
1
−
5
incenter wanted , <BIC = < IDC, cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral, and angle bisectors of
∠
B
A
D
\angle BAD
∠
B
A
D
and
∠
B
C
D
\angle BCD
∠
BC
D
meet at point
I
I
I
. Show that if
∠
B
I
C
=
∠
I
D
C
\angle BIC = \angle IDC
∠
B
I
C
=
∠
I
D
C
, then
I
I
I
is the incenter of triangle
A
B
D
ABD
A
B
D
.
exist 2 disjoint subsets of A with same sum of elements
Let
A
A
A
be the subset of
{
1
,
2
,
.
.
.
,
16
}
\{1, 2, ..., 16\}
{
1
,
2
,
...
,
16
}
that has
6
6
6
elements. Prove that there exist
2
2
2
subsets of
A
A
A
that are disjoint, and the sum of their elements are the same.
3
4
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