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National and Regional Contests
Ukraine Contests
Other Ukrainian Contests
Kyiv Mathematical Festival
2008 Kyiv Mathematical Festival
2008 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
5
1
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marking the squares on the chessboard-1
Some
m
m
m
squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest
m
m
m
for which it is possible to mark all squares after several such operations.
4
1
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points on two circles
Let
K
,
L
,
M
K,L,M
K
,
L
,
M
and
N
N
N
be the midpoints of sides
A
B
,
AB,
A
B
,
B
C
,
BC,
BC
,
C
D
CD
C
D
and
A
D
AD
A
D
of the convex quadrangle
A
B
C
D
.
ABCD.
A
BC
D
.
Is it possible that points
A
,
B
,
L
,
M
,
D
A,B,L,M,D
A
,
B
,
L
,
M
,
D
lie on the same circle and points
K
,
B
,
C
,
D
,
N
K,B,C,D,N
K
,
B
,
C
,
D
,
N
lie on the same circle?
3
1
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7 integers
Prove that among any 7 integers there exist three numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac is divisible by 7.
2
1
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Aladdin's set of coins
Aladdin has a set of coins with weights
1
,
2
,
…
,
20
1, 2, \ldots, 20
1
,
2
,
…
,
20
grams. He can ask Genie about any two coins from the set which one is heavier, but he should pay Genie some other coin from the set before. (So, with every question the set of coins becomes smaller.) Can Aladdin find two coins from the set with total weight at least
28
28
28
grams?
1
1
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quite easy equation
Find all positive integers
k
k
k
for which equation n^m\minus{}m^n\equal{}k has solution in positive integers.