MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2015 Kazakhstan National Olympiad
2015 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(6)
3
1
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Geometry concurrent
A rectangle is said to be
i
n
s
c
r
i
b
e
d
inscribed
in
scr
ib
e
d
in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
6
1
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Prove or disprove that
The quadrilateral
A
B
C
D
ABCD
A
BC
D
has an incircle of diameter
d
d
d
which touches
B
C
BC
BC
at
K
K
K
and touches
D
A
DA
D
A
at
L
L
L
. Is it always true that the harmonic mean of
A
B
AB
A
B
and
C
D
CD
C
D
is equal to
K
L
KL
K
L
if and only if the geometric mean of
A
B
AB
A
B
and
C
D
CD
C
D
is equal to
d
d
d
?
4
1
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perfect square
P
k
(
n
)
P_k(n)
P
k
(
n
)
is the product of all positive divisors of
n
n
n
that are divisible by
k
k
k
(the empty product is equal to
1
1
1
). Show that
P
1
(
n
)
P
2
(
n
)
⋯
P
n
(
n
)
P_1(n)P_2(n)\cdots P_n(n)
P
1
(
n
)
P
2
(
n
)
⋯
P
n
(
n
)
is a perfect square, for any positive integer
n
n
n
.
5
1
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find all possible permutations
Find all possible
{
x
1
,
x
2
,
.
.
.
x
n
}
\{ x_1,x_2,...x_n \}
{
x
1
,
x
2
,
...
x
n
}
permutations of
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n \}
{
1
,
2
,
...
,
n
}
so that when
1
≤
i
≤
n
−
2
1\le i \le n-2
1
≤
i
≤
n
−
2
then we have
x
i
<
x
i
+
2
x_i < x_{i+2}
x
i
<
x
i
+
2
and when
1
≤
i
≤
n
−
3
1 \le i \le n-3
1
≤
i
≤
n
−
3
then we have
x
i
<
x
i
+
3
x_i < x_{i+3}
x
i
<
x
i
+
3
. Here
n
≥
4
n \ge 4
n
≥
4
.
2
1
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From which NMO is this?
Solve in positive integers
x
y
y
x
=
(
x
+
y
)
z
x^yy^x=(x+y)^z
x
y
y
x
=
(
x
+
y
)
z
1
1
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Bounding the sum of squares of reciprocals
Prove that
1
2
2
+
1
3
2
+
⋯
+
1
(
n
+
1
)
2
<
n
⋅
(
1
−
1
2
n
)
.
\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).
2
2
1
+
3
2
1
+
⋯
+
(
n
+
1
)
2
1
<
n
⋅
(
1
−
n
2
1
)
.