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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2005 Ukraine National Mathematical Olympiad
2005 Ukraine National Mathematical Olympiad
Part of
Ukraine National Mathematical Olympiad
Subcontests
(8)
8
3
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construct a point
Let
A
B
AB
A
B
and
C
D
CD
C
D
be two disjoint chords of a circle. A point
E
E
E
, distinct from
A
A
A
and
B
B
B
, is taken on the chord
A
B
AB
A
B
. Consider the arc
A
B
AB
A
B
not containing
C
C
C
and
D
D
D
. Using a ruler and a compass, construct a point
F
F
F
on this arc such that \frac{PE}{EQ}\equal{}\frac{1}{2}, where
P
P
P
and
Q
Q
Q
are the intersection points of
A
B
AB
A
B
with the segments
F
C
FC
FC
and
F
D
FD
F
D
, respectively.
points
On the plane are marked
2005
2005
2005
points, no three of which are collinear. A line is drawn through any two of the points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn lines separating them is even. (Two points are separated by a line if they lie in different open half-planes determined by the line).
colored points
In space are marked
2005
2005
2005
points, no four of which are in the same plane. A plane is drawn through any three points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn planes separating them is odd. (Two points are separated by a plane if they lie in different open half-spaces determined by the plane).
7
3
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game
Under the rules of the "Sea battle" game (no two ships may have a common point), can the following sets of rectangular ships be arranged on a
10
×
10
10 \times 10
10
×
10
square board:
(
a
)
(a)
(
a
)
two ships
1
×
4
1 \times 4
1
×
4
,
4
4
4
ships
1
×
3
1 \times 3
1
×
3
,
6
6
6
ships
1
×
2
1 \times 2
1
×
2
, and
8
8
8
ships
1
×
1
1 \times 1
1
×
1
;
(
b
)
(b)
(
b
)
two ships
1
×
4
1 \times 4
1
×
4
,
4
4
4
ships
1
×
3
1 \times 3
1
×
3
,
6
6
6
ships
1
×
2
1 \times 2
1
×
2
,
6
6
6
ships
1
×
1
1 \times 1
1
×
1
, and
1
1
1
ship
2
×
2
2 \times 2
2
×
2
;
(
c
)
(c)
(
c
)
two ships
1
×
4
1 \times 4
1
×
4
,
4
4
4
ships
1
×
3
1 \times 3
1
×
3
,
6
6
6
ships
1
×
2
1 \times 2
1
×
2
,
4
4
4
ships
1
×
1
1 \times 1
1
×
1
, and
2
2
2
ships
2
×
2
2 \times 2
2
×
2
?
find the measure of an angle
A point
M
M
M
is taken on the perpendicular bisector of the side
A
C
AC
A
C
of an acute-angled triangle
A
B
C
ABC
A
BC
so that
M
M
M
and
B
B
B
are on the same side of
A
C
AC
A
C
. If \angle BAC\equal{}\angle MCB and \angle ABC\plus{}\angle MBC\equal{}180^{\circ}, find
∠
B
A
C
.
\angle BAC.
∠
B
A
C
.
divisibility
Prove that for any integers
n
≥
2
n \ge 2
n
≥
2
there is a set
A
n
A_n
A
n
of
n
n
n
distinct positive integers such that for any two distinct elements i,j \in A_n, |i\minus{}j| divides i^2\plus{}j^2.
6
4
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5
4
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3
4
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1
4
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2
4
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4
4
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