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Tournament Of Towns
2009 Tournament Of Towns
2009 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(7)
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8
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4
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3
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Infinite chessboard - TT 2009 Junior-A6
On an in finite chessboard are placed
2009
n
×
n
2009 \ n \times n
2009
n
×
n
cardboard pieces such that each of them covers exactly
n
2
n^2
n
2
cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least
n
2
.
n^2.
n
2
.
(9 points)
2009 ToT Spring Junior A P7 sum of triangle areas
Angle
C
C
C
of an isosceles triangle
A
B
C
ABC
A
BC
equals
12
0
o
120^o
12
0
o
. Each of two rays emitting from vertex
C
C
C
(inwards the triangle) meets
A
B
AB
A
B
at some point (
P
i
P_i
P
i
) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle
A
B
C
ABC
A
BC
at point
Q
i
Q_i
Q
i
(
i
=
1
,
2
i = 1,2
i
=
1
,
2
). Given that angle between the rays equals
6
0
o
60^o
6
0
o
, prove that area of triangle
P
1
C
P
2
P_1CP_2
P
1
C
P
2
equals the sum of areas of triangles
A
Q
1
P
1
AQ_1P_1
A
Q
1
P
1
and
B
Q
2
P
2
BQ_2P_2
B
Q
2
P
2
(
A
P
1
<
A
P
2
AP_1 < AP_2
A
P
1
<
A
P
2
).
2009 ToT Spring Senior A P6 marking points in circle game
An integer
n
>
1
n > 1
n
>
1
is given. Two players in turns mark points on a circle. First Player uses red color while Second Player uses blue color. The game is over when each player marks
n
n
n
points. Then each player nds the arc of maximal length with ends of his color, which does not contain any other marked points. A player wins if his arc is longer (if the lengths are equal, or both players have no such arcs, the game ends in a draw). Which player has a winning strategy?
5
7
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8
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2
7
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1
6
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