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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2003 Serbia Team Selection Test
2003 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
2
1
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AM : BM : CM = AN : BN : CN
Let M and N be the distinct points in the plane of the triangle ABC such that AM : BM : CM = AN : BN : CN. Prove that the line MN contains the circumcenter of △ABC.
1
1
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factoring p(x)-x
If
p
(
x
)
p(x)
p
(
x
)
is a polynomial, denote by
p
n
(
x
)
p^n(x)
p
n
(
x
)
the polynomial
p
(
p
(
.
.
.
(
p
(
x
)
)
.
.
)
p(p(...(p(x))..)
p
(
p
(
...
(
p
(
x
))
..
)
, where
p
p
p
is iterated
n
n
n
times. Prove that the polynomial p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x) is divisible by p(x)\minus{}x
3
1
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arranging the convex n-gon
Each edge and each diagonal of the convex
n
n
n
-gon
(
n
≥
3
)
(n\geq 3)
(
n
≥
3
)
is colored in red or blue. Prove that the vertices of the
n
n
n
-gon can be labeled as
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
in such a way that one of the following two conditions is satisfied: (a) all segments A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1 are of the same colour. (b) there exists a number
k
,
1
<
k
<
n
k, 1<k<n
k
,
1
<
k
<
n
such that the segments A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k are blue, and the segments A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1 are red.