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Problems
Contests
International Contests
Tuymaada Olympiad
2007 Tuymaada Olympiad
2007 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(4)
4
2
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determine maximum real k
Determine maximum real
k
k
k
such that there exist a set
X
X
X
and its subsets
Y
1
Y_{1}
Y
1
,
Y
2
Y_{2}
Y
2
,
.
.
.
...
...
,
Y
31
Y_{31}
Y
31
satisfying the following conditions: (1) for every two elements of
X
X
X
there is an index
i
i
i
such that
Y
i
Y_{i}
Y
i
contains neither of these elements; (2) if any non-negative numbers
α
i
\alpha_{i}
α
i
are assigned to the subsets
Y
i
Y_{i}
Y
i
and
α
1
+
⋯
+
α
31
=
1
\alpha_{1}+\dots+\alpha_{31}=1
α
1
+
⋯
+
α
31
=
1
then there is an element
x
∈
X
x\in X
x
∈
X
such that the sum of
α
i
\alpha_{i}
α
i
corresponding to all the subsets
Y
i
Y_{i}
Y
i
that contain
x
x
x
is at least
k
k
k
.
prove that there exists a positive $c$...
Prove that there exists a positive
c
c
c
such that for every positive integer
N
N
N
among any
N
N
N
positive integers not exceeding
2
N
2N
2
N
there are two numbers whose greatest common divisor is greater than
c
N
cN
c
N
.
3
2
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altitudes and circles tangent to the circumcircle
A
A
1
AA_{1}
A
A
1
,
B
B
1
BB_{1}
B
B
1
,
C
C
1
CC_{1}
C
C
1
are altitudes of an acute triangle
A
B
C
ABC
A
BC
. A circle passing through
A
1
A_{1}
A
1
and
B
1
B_{1}
B
1
touches the arc
A
B
AB
A
B
of its circumcircle at
C
2
C_{2}
C
2
. The points
A
2
A_{2}
A
2
,
B
2
B_{2}
B
2
are defined similarly. Prove that the lines
A
A
2
AA_{2}
A
A
2
,
B
B
2
BB_{2}
B
B
2
,
C
C
2
CC_{2}
C
C
2
are concurrent.
Sasha playing with knights...
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a
14
×
16
14\times 16
14
×
16
rectangle. What maximum number of knights can this rectangle contain?
2
2
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two polynomials of degree 100
Two polynomials
f
(
x
)
=
a
100
x
100
+
a
99
x
99
+
⋯
+
a
1
x
+
a
0
f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}
f
(
x
)
=
a
100
x
100
+
a
99
x
99
+
⋯
+
a
1
x
+
a
0
and
g
(
x
)
=
b
100
x
100
+
b
99
x
99
+
⋯
+
b
1
x
+
b
0
g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}
g
(
x
)
=
b
100
x
100
+
b
99
x
99
+
⋯
+
b
1
x
+
b
0
of degree
100
100
100
differ from each other by a permutation of coefficients. It is known that
a
i
≠
b
i
a_{i}\ne b_{i}
a
i
=
b
i
for
i
=
0
,
1
,
2
,
…
,
100
i=0, 1, 2, \dots, 100
i
=
0
,
1
,
2
,
…
,
100
. Is it possible that
f
(
x
)
≥
g
(
x
)
f(x)\geq g(x)
f
(
x
)
≥
g
(
x
)
for all real
x
x
x
?
slicing problem with a right angle in the end
Point
D
D
D
is chosen on the side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
. Point
L
L
L
inside the triangle
A
B
C
ABC
A
BC
is such that
B
D
=
L
D
BD=LD
B
D
=
L
D
and
∠
L
A
B
=
∠
L
C
A
=
∠
D
C
B
\angle LAB=\angle LCA=\angle DCB
∠
L
A
B
=
∠
L
C
A
=
∠
D
CB
. It is known that
∠
A
L
D
+
∠
A
B
C
=
18
0
∘
\angle ALD+\angle ABC=180^\circ
∠
A
L
D
+
∠
A
BC
=
18
0
∘
. Prove that
∠
B
L
C
=
9
0
∘
\angle BLC=90^\circ
∠
B
L
C
=
9
0
∘
.
1
2
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more like trivial colouring exercise
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
classic divisibility..
Positive integers
a
<
b
a<b
a
<
b
are given. Prove that among every
b
b
b
consecutive positive integers there are two numbers whose product is divisible by
a
b
ab
ab
.