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Kazakhstan Contests
Kazakhstan National Olympiad
2013 Kazakhstan National Olympiad
2013 Kazakhstan National Olympiad
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Kazakhstan National Olympiad
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KMN and PQR are tangent at a fixed point
Let
A
B
C
D
ABCD
A
BC
D
be cyclic quadrilateral. Let
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
R
R
R
, and let
A
B
AB
A
B
and
C
D
CD
C
D
intersect at
K
K
K
. Let
M
M
M
and
N
N
N
are points on
A
B
AB
A
B
and
C
D
CD
C
D
such that
A
M
M
B
=
C
N
N
D
\frac{AM}{MB}=\frac{CN}{ND}
MB
A
M
=
N
D
CN
. Let
P
P
P
and
Q
Q
Q
be the intersections of
M
N
MN
MN
with the diagonals of
A
B
C
D
ABCD
A
BC
D
. Prove that circumcircles of triangles
K
M
N
KMN
K
MN
and
P
Q
R
PQR
PQR
are tangent at a fixed point.
Kazaakhstan National Olympiad 2013,Problem 6,10th grade
How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?
Sequence ...
Consider the following sequence :
a
1
=
1
;
a
n
=
a
[
n
2
]
2
+
a
[
n
3
]
3
+
…
+
a
[
n
n
]
n
a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}
a
1
=
1
;
a
n
=
2
a
[
2
n
]
+
3
a
[
3
n
]
+
…
+
n
a
[
n
n
]
. Prove that
a
2
n
<
2
∗
a
n
(
∀
n
∈
N
)
a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})
a
2
n
<
2
∗
a
n
(
∀
n
∈
N
)