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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1974 Bulgaria National Olympiad
1974 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
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inequality in triangular pyramid
In triangle pyramid
M
A
B
C
MABC
M
A
BC
at least two of the plane angles next to the edge
M
M
M
are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
8
a
1
b
1
c
1
≤
a
2
a
1
+
b
2
b
1
+
c
2
c
1
8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1
8
a
1
b
1
c
1
≤
a
2
a
1
+
b
2
b
1
+
c
2
c
1
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are sides of triangle
A
B
C
ABC
A
BC
and
a
1
,
b
1
,
c
1
a_1,b_1,c_1
a
1
,
b
1
,
c
1
are edges crossed respectively with
a
,
b
,
c
a,b,c
a
,
b
,
c
.V. Petkov
Problem 5
1
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maximize triangle formed by feet from point in triangle
Find all point
M
M
M
lying into given acute-angled triangle
A
B
C
ABC
A
BC
and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from
M
M
M
to the lines
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
is maximal.H. Lesov
Problem 4
1
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placing shapes on chessboard, 8x8
Find the maximal count of shapes that can be placed over a chessboard with size
8
×
8
8\times8
8
×
8
in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel
A
1
−
H
8
A1-H8
A
1
−
H
8
(
A
1
A1
A
1
is the lowest-bottom left corner of the chessboard,
H
8
H8
H
8
is the highest-upper right corner of the chessboard).V. Chukanov
Problem 3
1
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a^2+b^2+c^2≥p(ab+bc), and the four variable variation
(a) Find all real numbers
p
p
p
for which the inequality
x
1
2
+
x
2
2
+
x
3
2
≥
p
(
x
1
x
2
+
x
2
x
3
)
x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)
x
1
2
+
x
2
2
+
x
3
2
≥
p
(
x
1
x
2
+
x
2
x
3
)
is true for all real numbers
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
. (b) Find all real numbers
q
q
q
for which the inequality
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
≥
q
(
x
1
x
2
+
x
2
x
3
+
x
3
x
4
)
x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
≥
q
(
x
1
x
2
+
x
2
x
3
+
x
3
x
4
)
is true for all real numbers
x
1
,
x
2
,
x
3
,
x
4
x_1,x_2,x_3,x_4
x
1
,
x
2
,
x
3
,
x
4
.I. Tonov
Problem 1
1
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people sitting on circular table, shifting σ(n) places
Find all natural numbers n with the following property: there exists a permutation
(
i
1
,
i
2
,
…
,
i
n
)
(i_1,i_2,\ldots,i_n)
(
i
1
,
i
2
,
…
,
i
n
)
of the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
such that, if on the circular table there are
n
n
n
people seated and for all
k
=
1
,
2
,
…
,
n
k=1,2,\ldots,n
k
=
1
,
2
,
…
,
n
the
k
k
k
-th person is moving
i
n
i_n
i
n
places in the right, all people will sit on different places.V. Drenski
Problem 2
1
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P(m^n)+Q([0,k]) is composite for infinitely many k
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be non-constant polynomials with integer positive coefficients,
m
m
m
and
n
n
n
are given natural numbers. Prove that there exists infinitely many natural numbers
k
k
k
for which the numbers
f
(
m
n
)
+
g
(
0
)
,
f
(
m
n
)
+
g
(
1
)
,
…
,
f
(
m
n
)
+
g
(
k
)
f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)
f
(
m
n
)
+
g
(
0
)
,
f
(
m
n
)
+
g
(
1
)
,
…
,
f
(
m
n
)
+
g
(
k
)
are composite.I. Tonov