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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2008 Ukraine Team Selection Test
2008 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(8)
12
1
Hide problems
Polynomial has at most one real root
Prove that for all natural
m
m
m
,
n
n
n
polynomial \sum_{i \equal{} 0}^{m}\binom{n\plus{}i}{n}\cdot x^i has at most one real root.
11
1
Hide problems
Convex pentagon and areas of triangles
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be convex pentagon such that S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB). Prove that there is a point
M
M
M
inside pentagon such that S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA).
9
1
Hide problems
Triangle with an arbitrary point inside
Given
△
A
B
C
\triangle ABC
△
A
BC
with point
D
D
D
inside. Let A_0\equal{}AD\cap BC, B_0\equal{}BD\cap AC, C_0 \equal{}CD\cap AB and
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
are midpoints of
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
,
A
D
AD
A
D
,
B
D
BD
B
D
,
C
D
CD
C
D
respectively. Two lines parallel to
A
1
A
2
A_1A_2
A
1
A
2
and
C
1
C
2
C_1C_2
C
1
C
2
and passes through point
B
0
B_0
B
0
intersects
B
1
B
2
B_1B_2
B
1
B
2
in points
A
3
A_3
A
3
and
C
3
C_3
C
3
respectively. Prove that \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}.
7
1
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Graph
There is graph
G
0
G_0
G
0
on vertices
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
. Graph G_{n \plus{} 1} on vertices
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
is constructed by the rule:
A
i
A_i
A
i
and
A
j
A_j
A
j
are joined only if in graph
G
n
G_n
G
n
there is a vertices
A
k
≠
A
i
,
A
j
A_k\neq A_i, A_j
A
k
=
A
i
,
A
j
such that
A
k
A_k
A
k
is joined with both
A
i
A_i
A
i
and
A
j
A_j
A
j
. Prove that the sequence
{
G
n
}
n
∈
N
\{G_n\}_{n\in\mathbb{N}}
{
G
n
}
n
∈
N
is periodic after some term with period
T
≤
2
n
T \le 2^n
T
≤
2
n
.
6
1
Hide problems
there exist infinitely many pairs of natural number
Prove that there exist infinitely many pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of natural numbers not equal to
1
1
1
such that b^b \plus{}a is divisible by a^a \plus{}b.
4
1
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Two internal tangent circles
Two circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
tangents internally in point
P
P
P
. On their common tangent points
A
A
A
,
B
B
B
are chosen such that
P
P
P
lies between
A
A
A
and
B
B
B
. Let
C
C
C
and
D
D
D
be the intersection points of tangent from
A
A
A
to
ω
1
\omega_1
ω
1
, tangent from
B
B
B
to
ω
2
\omega_2
ω
2
and tangent from
A
A
A
to
ω
2
\omega_2
ω
2
, tangent from
B
B
B
to
ω
1
\omega_1
ω
1
, respectively. Prove that CA \plus{} CB \equal{} DA \plus{} DB.
2
1
Hide problems
Find the greatest possible number of symbols
There is a row that consists of digits from
0
0
0
to
9
9
9
and Ukrainian letters (there are
33
33
33
of them) with following properties: there aren’t two distinct digits or letters
a
i
a_i
a
i
,
a
j
a_j
a
j
such that
a
i
>
a
j
a_i > a_j
a
i
>
a
j
and
i
<
j
i < j
i
<
j
(if
a
i
a_i
a
i
,
a
j
a_j
a
j
are letters
a
i
>
a
j
a_i > a_j
a
i
>
a
j
means that
a
i
a_i
a
i
has greater then
a
j
a_j
a
j
position in alphabet) and there aren’t two equal consecutive symbols or two equal symbols having exactly one symbol between them. Find the greatest possible number of symbols in such row.
3
1
Hide problems
Ukraine 2008
For positive
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
prove that (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)