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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2021 Bulgaria National Olympiad
2021 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(5)
6
1
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geometry around incenter and midpoint of arc ACB
Point
S
S
S
is the midpoint of arc
A
C
B
ACB
A
CB
of the circumscribed circle
k
k
k
around triangle
A
B
C
ABC
A
BC
with
A
C
>
B
C
AC>BC
A
C
>
BC
. Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
. Line
S
I
SI
S
I
intersects
k
k
k
again at point
T
T
T
. Let
D
D
D
be the reflection of
I
I
I
across
T
T
T
and
M
M
M
be the midpoint of side
A
B
AB
A
B
. Line
I
M
IM
I
M
intersects the line through
D
D
D
, parallel to
A
B
AB
A
B
, at point
E
E
E
. Prove that
A
E
=
B
D
AE=BD
A
E
=
B
D
.
4
1
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divisibility in arithmetic sequences
Two infinite arithmetic sequences with positive integers are given:
a
1
<
a
2
<
a
3
<
⋯
;
b
1
<
b
2
<
b
3
<
⋯
a_1<a_2<a_3<\cdots ; b_1<b_2<b_3<\cdots
a
1
<
a
2
<
a
3
<
⋯
;
b
1
<
b
2
<
b
3
<
⋯
It is known that there are infinitely many pairs of positive integers
(
i
,
j
)
(i,j)
(
i
,
j
)
for which
i
≤
j
≤
i
+
2021
i\leq j\leq i+2021
i
≤
j
≤
i
+
2021
and
a
i
a_i
a
i
divides
b
j
b_j
b
j
. Prove that for every positive integer
i
i
i
there exists a positive integer
j
j
j
such that
a
i
a_i
a
i
divides
b
j
b_j
b
j
.
2
1
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point on the altitude
A point
T
T
T
is given on the altitude through point
C
C
C
in the acute triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
, such that
∡
T
B
A
=
∡
A
C
B
\measuredangle TBA=\measuredangle ACB
∡
TB
A
=
∡
A
CB
. If the line
C
O
CO
CO
intersects side
A
B
AB
A
B
at point
K
K
K
, prove that the perpendicular bisector of
A
B
AB
A
B
, the altitude through
A
A
A
and the segment
K
T
KT
K
T
are concurrent.
1
1
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4n crossroads on boulevards
A city has
4
4
4
horizontal and
n
≥
3
n\geq3
n
≥
3
vertical boulevards which intersect at
4
n
4n
4
n
crossroads. The crossroads divide every horizontal boulevard into
n
−
1
n-1
n
−
1
streets and every vertical boulevard into
3
3
3
streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city doesn't have a closed path(this means that starting from any street and going only through open crossroads without turning back you can't return to the same street).
a
)
a)
a
)
Prove that exactly
n
n
n
crossroads are closed.
b
)
b)
b
)
Prove that if from any street you can go to any other street and none of the
4
4
4
corner crossroads are closed then exactly
3
3
3
crossroads on the border are closed(A crossroad is on the border if it lies either on the first or fourth horizontal boulevard, or on the first or the n-th vertical boulevard).
3
1
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f(f(x) + y)f(x) = f(xy + 1)
Find all
f
:
R
+
→
R
+
f:R^+ \rightarrow R^+
f
:
R
+
→
R
+
such that
f
(
f
(
x
)
+
y
)
f
(
x
)
=
f
(
x
y
+
1
)
∀
x
,
y
∈
R
+
f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+
f
(
f
(
x
)
+
y
)
f
(
x
)
=
f
(
x
y
+
1
)
∀
x
,
y
∈
R
+
@below: https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problemsFeel free to start individual threads for the problems as usual