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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2019 Bulgaria National Olympiad
2019 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
6
1
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A geometry problem
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be an inscribed hexagon with
A
B
.
C
D
.
E
F
=
B
C
.
D
E
.
F
A
AB.CD.EF=BC.DE.FA
A
B
.
C
D
.
EF
=
BC
.
D
E
.
F
A
Let
B
1
B_1
B
1
be the reflection point of
B
B
B
with respect to
A
C
AC
A
C
and
D
1
D_1
D
1
be the reflection point of
D
D
D
with respect to
C
E
,
CE,
CE
,
and finally let
F
1
F_1
F
1
be the reflection point of
F
F
F
with respect to
A
E
.
AE.
A
E
.
Prove that
△
B
1
D
1
F
1
∼
B
D
F
.
\triangle B_1D_1F_1\sim BDF.
△
B
1
D
1
F
1
∼
B
D
F
.
5
1
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A combinatorial geometry
Let
P
P
P
be a
2019
−
2019-
2019
−
gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of
P
P
P
a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)
4
1
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An easy number theory
Determine all positive integers
d
,
d,
d
,
such that there exists an integer
k
≥
3
,
k\geq 3,
k
≥
3
,
such that One can arrange the numbers
d
,
2
d
,
…
,
k
d
d,2d,\ldots,kd
d
,
2
d
,
…
,
k
d
in a row, such that the sum of every two consecutive of them is a perfect square.
3
1
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A sequence of integers
Find all real numbers
a
,
a,
a
,
which satisfy the following condition:For every sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
of pairwise different positive integers, for which the inequality
a
n
≤
a
n
a_n\leq an
a
n
≤
an
holds for every positive integer
n
,
n,
n
,
there exist infinitely many numbers in the sequence with sum of their digits in base
4038
,
4038,
4038
,
which is not divisible by
2019.
2019.
2019.
2
1
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Nice geometry
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocenter
H
H
H
and circumcenter
O
.
O.
O
.
Let the intersection points of the perpendicular bisector of
C
H
CH
C
H
with
A
C
AC
A
C
and
B
C
BC
BC
be
X
X
X
and
Y
Y
Y
respectively. Lines
X
O
XO
XO
and
Y
O
YO
Y
O
cut
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
respectively. If
X
P
+
Y
Q
=
A
B
+
X
Y
,
XP+YQ=AB+XY,
XP
+
Y
Q
=
A
B
+
X
Y
,
determine
∡
O
H
C
.
\measuredangle OHC.
∡
O
H
C
.
1
1
Hide problems
One variable inequality
Let
f
(
x
)
=
x
2
+
b
x
+
1
,
f(x)=x^2+bx+1,
f
(
x
)
=
x
2
+
b
x
+
1
,
where
b
b
b
is a real number. Find the number of integer solutions to the inequality
f
(
f
(
x
)
+
x
)
<
0.
f(f(x)+x)<0.
f
(
f
(
x
)
+
x
)
<
0.