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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2016 Bulgaria National Olympiad
2016 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(5)
Problem 6
1
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Bulgarian National Mathematical Olympiad 2016, Problem 6
Let
n
n
n
be positive integer.A square
A
A
A
of side length
n
n
n
is divided by
n
2
n^2
n
2
unit squares. All unit squares are painted in
n
n
n
distinct colors such that each color appears exactly
n
n
n
times. Prove that there exists a positive integer
N
N
N
, such that for any
n
>
N
n>N
n
>
N
the following is true: There exists a square
B
B
B
of side length
n
\sqrt{n}
n
and side parallel to the sides of
A
A
A
such that
B
B
B
contains completely cells of
4
4
4
distinct colors.
Problem 5
1
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Bulgarian National Mathematical Olympiad 2016, Problem 5
Let
△
A
B
C
\triangle {ABC}
△
A
BC
be isosceles triangle with
A
C
=
B
C
AC=BC
A
C
=
BC
. The point
D
D
D
lies on the extension of
A
C
AC
A
C
beyond
C
C
C
and is that
A
C
>
C
D
AC>CD
A
C
>
C
D
. The angular bisector of
∠
B
C
D
\angle BCD
∠
BC
D
intersects
B
D
BD
B
D
at point
N
N
N
and let
M
M
M
be the midpoint of
B
D
BD
B
D
. The tangent at
M
M
M
to the circumcircle of triangle
A
M
D
AMD
A
M
D
intersects the side
B
C
BC
BC
at point
P
P
P
. Prove that points
A
,
P
,
M
A,P,M
A
,
P
,
M
and
N
N
N
lie on a circle.
Problem 4
1
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Bulgaria National Mathematical Olympiad 2016, Problem 4
Determine whether there exist a positive integer
n
<
1
0
9
n<10^9
n
<
1
0
9
, such that
n
n
n
can be expressed as a sum of three squares of positive integers by more than
1000
1000
1000
distinct ways?
Problem 2
1
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Bulgarian National Mathematical Olympiad 2016 ,problem 2
At a mathematical competition
n
n
n
students work on
6
6
6
problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is
0
0
0
or
2
2
2
. Find the maximum possible value of
n
n
n
.
Problem 3
1
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Another nice Inequality
For
a
,
b
,
c
,
d
>
0
a,b,c,d>0
a
,
b
,
c
,
d
>
0
prove that
a
+
a
b
+
a
b
c
3
+
a
b
c
d
4
4
≤
a
.
a
+
b
2
.
a
+
b
+
c
3
.
a
+
b
+
c
+
d
4
4
\frac {a+\sqrt{ab}+\sqrt[3]{abc}+\sqrt[4]{abcd}}{4} \leq \sqrt[4]{a.\frac{a+b}{2}.\frac{a+b+c}{3}.\frac{a+b+c+d}{4}}
4
a
+
ab
+
3
ab
c
+
4
ab
c
d
≤
4
a
.
2
a
+
b
.
3
a
+
b
+
c
.
4
a
+
b
+
c
+
d