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Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2011 Kyrgyzstan National Olympiad
2011 Kyrgyzstan National Olympiad
Part of
Kyrgyzstan National Olympiad
Subcontests
(8)
3
1
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Square difference(KgNM2011)
Given positive numbers
a
1
,
a
2
,
.
.
.
,
a
n
{a_1},{a_2},...,{a_n}
a
1
,
a
2
,
...
,
a
n
with
a
1
+
a
2
+
.
.
.
+
a
n
=
1
{a_1} + {a_2} + ... + {a_n} = 1
a
1
+
a
2
+
...
+
a
n
=
1
. Prove that
(
1
a
1
2
−
1
)
(
1
a
2
2
−
1
)
.
.
.
(
1
a
n
2
−
1
)
⩾
(
n
2
−
1
)
n
\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}
(
a
1
2
1
−
1
)
(
a
2
2
1
−
1
)
...
(
a
n
2
1
−
1
)
⩾
(
n
2
−
1
)
n
.
1
1
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Chord problem(KgNM2011)
For a given chord
M
N
MN
MN
of a circle discussed the triangle
A
B
C
ABC
A
BC
, whose base is the diameter
A
B
AB
A
B
of this circle,which do not intersect the
M
N
MN
MN
, and the sides
A
C
AC
A
C
and
B
C
BC
BC
pass through the ends of
M
M
M
and
N
N
N
of the chord
M
N
MN
MN
. Prove that the heights of all such triangles
A
B
C
ABC
A
BC
drawn from the vertex
C
C
C
to the side
A
B
AB
A
B
, intersect at one point.
2
1
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regular n-gon(KgNM2011)
In a convex
n
n
n
-gon all angles are equal from a certain point, located inside the
n
n
n
-gon, all its sides are seen under equal angles. Can we conclude that this
n
n
n
-gon is regular?
5
1
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Geo Problem(KgNM2011)
Points
M
M
M
and
N
N
N
are chosen on sides
A
B
AB
A
B
and
B
C
BC
BC
,respectively, in a triangle
A
B
C
ABC
A
BC
, such that point
O
O
O
is interserction of lines
C
M
CM
CM
and
A
N
AN
A
N
. Given that
A
M
+
A
N
=
C
M
+
C
N
AM+AN=CM+CN
A
M
+
A
N
=
CM
+
CN
. Prove that
A
O
+
A
B
=
C
O
+
C
B
AO+AB=CO+CB
A
O
+
A
B
=
CO
+
CB
.
6
1
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Distance problem(KgNM2011)
a) Among the
21
21
21
pairwise distances between the
7
7
7
points of the plane, prove that one and the same number occurs not more than
12
12
12
times.b) Find a maximum number of times may meet the same number among the
15
15
15
pairwise distances between
6
6
6
points of the plane.
7
1
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Difference problem(KgNM2011)
Given that
g
(
n
)
=
1
2
+
1
3
+
1
.
.
.
+
1
n
−
1
g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}
g
(
n
)
=
2
+
3
+
...
+
n
−
1
1
1
1
1
and
k
(
n
)
=
1
2
+
1
3
+
1
.
.
.
+
1
n
−
1
+
1
n
k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}
k
(
n
)
=
2
+
3
+
...
+
n
−
1
+
n
1
1
1
1
1
, for natural
n
n
n
. Prove that
∣
g
(
n
)
−
k
(
n
)
∣
≤
1
(
n
−
1
)
!
n
!
\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}
∣
g
(
n
)
−
k
(
n
)
∣
≤
(
n
−
1
)!
n
!
1
.
4
1
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Algebraic ineq(KgNM2011)
Given equation
a
5
−
a
3
+
a
=
2
{a^5} - {a^3} + a = 2
a
5
−
a
3
+
a
=
2
, with real
a
a
a
. Prove that
3
<
a
6
<
4
3 < {a^6} < 4
3
<
a
6
<
4
.
8
1
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Sequence problem (KgNM2011)
Given a sequence
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
of real numbers with
x
n
+
1
3
=
x
n
3
−
3
x
n
2
+
3
x
n
{x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}
x
n
+
1
3
=
x
n
3
−
3
x
n
2
+
3
x
n
, where
(
n
=
1
,
2
,
3
,
.
.
.
)
(n=1,2,3,...)
(
n
=
1
,
2
,
3
,
...
)
. What must be value of
x
1
x_1
x
1
, so that
x
100
x_{100}
x
100
and
x
1000
x_{1000}
x
1000
becomes equal?