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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2022 Bulgaria National Olympiad
2022 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
6
1
Hide problems
Differences between elements of a set
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer. The sets
A
1
,
A
2
,
…
,
A
n
A_{1},A_{2},\ldots, A_{n}
A
1
,
A
2
,
…
,
A
n
and
B
1
,
B
2
,
…
,
B
n
B_{1},B_{2},\ldots, B_{n}
B
1
,
B
2
,
…
,
B
n
of positive integers are such that
A
i
∩
B
j
A_{i}\cap B_{j}
A
i
∩
B
j
is non-empty
∀
i
,
j
∈
{
1
,
2
,
…
,
n
}
\forall i,j\in\{1,2,\ldots ,n\}
∀
i
,
j
∈
{
1
,
2
,
…
,
n
}
and A_{i}\cap A_{j}=\o, B_{i}\cap B_{j}=\o
∀
i
≠
j
∈
{
1
,
2
,
…
,
n
}
\forall i\neq j\in \{1,2,\ldots, n\}
∀
i
=
j
∈
{
1
,
2
,
…
,
n
}
. We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.
5
1
Hide problems
Number theory hidden behind a geometrical statement
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
4
AB=4
A
B
=
4
,
B
C
=
C
A
=
6
BC=CA=6
BC
=
C
A
=
6
. On the segment
A
B
AB
A
B
consecutively lie points
X
1
,
X
2
,
X
3
,
…
X_{1},X_{2},X_{3},\ldots
X
1
,
X
2
,
X
3
,
…
such that the lengths of the segments
A
X
1
,
X
1
X
2
,
X
2
X
3
,
…
AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots
A
X
1
,
X
1
X
2
,
X
2
X
3
,
…
form an infinite geometric progression with starting value
3
3
3
and common ratio
1
4
\frac{1}{4}
4
1
. On the segment
C
B
CB
CB
consecutively lie points
Y
1
,
Y
2
,
Y
3
,
…
Y_{1},Y_{2},Y_{3},\ldots
Y
1
,
Y
2
,
Y
3
,
…
such that the lengths of the segments
C
Y
1
,
Y
1
Y
2
,
Y
2
Y
3
,
…
CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots
C
Y
1
,
Y
1
Y
2
,
Y
2
Y
3
,
…
form an infinite geometric progression with starting value
3
3
3
and common ratio
1
2
\frac{1}{2}
2
1
. On the segment
A
C
AC
A
C
consecutively lie points
Z
1
,
Z
2
,
Z
3
,
…
Z_{1},Z_{2},Z_{3},\ldots
Z
1
,
Z
2
,
Z
3
,
…
such that the lengths of the segments
A
Z
1
,
Z
1
Z
2
,
Z
2
Z
3
,
…
AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots
A
Z
1
,
Z
1
Z
2
,
Z
2
Z
3
,
…
form an infinite geometric progression with starting value
3
3
3
and common ratio
1
2
\frac{1}{2}
2
1
. Find all triplets of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that the segments
A
Y
a
AY_{a}
A
Y
a
,
B
Z
b
BZ_{b}
B
Z
b
and
C
X
c
CX_{c}
C
X
c
are concurrent.
4
1
Hide problems
Cyclic system of equations
Let
n
≥
4
n\geq 4
n
≥
4
be a positive integer and
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
be real numbers such that
x
n
+
1
=
x
1
x_{n+1}=x_{1}
x
n
+
1
=
x
1
and
x
n
+
2
=
x
2
x_{n+2}=x_{2}
x
n
+
2
=
x
2
. If there exists an
a
>
0
a>0
a
>
0
such that x_{i}^2=a+x_{i+1}x_{i+2} \forall 1\leq i\leq n then prove that at least
2
2
2
of the numbers
x
1
,
x
2
,
…
,
x
n
x_{1},x_{2},\ldots ,x_{n}
x
1
,
x
2
,
…
,
x
n
are negative.
3
1
Hide problems
Can a number not be coprime to many consecutive integers
Let
x
>
y
>
2022
x>y>2022
x
>
y
>
2022
be positive integers such that
x
y
+
x
+
y
xy+x+y
x
y
+
x
+
y
is a perfect square. Is it possible for every positive integer
z
z
z
from the interval
[
x
+
3
y
+
1
,
3
x
+
y
+
1
]
[x+3y+1,3x+y+1]
[
x
+
3
y
+
1
,
3
x
+
y
+
1
]
the numbers
x
+
y
+
z
x+y+z
x
+
y
+
z
and
x
2
+
x
y
+
y
2
x^2+xy+y^2
x
2
+
x
y
+
y
2
not to be coprime?
2
1
Hide problems
Cyclic quads imply tangency
Let
A
B
C
ABC
A
BC
be an acute triangle and
M
M
M
be the midpoint of
A
B
AB
A
B
. A circle through the points
B
B
B
and
C
C
C
intersects the segments
C
M
CM
CM
and
B
M
BM
BM
at points
P
P
P
and
Q
Q
Q
respectively. Point
K
K
K
is symmetric to
P
P
P
with respect to point
M
M
M
. The circumcircles of
△
A
K
M
\triangle AKM
△
A
K
M
and
△
C
Q
M
\triangle CQM
△
CQM
intersect for the second time at
X
X
X
. The circumcircles of
△
A
M
C
\triangle AMC
△
A
MC
and
△
K
M
Q
\triangle KMQ
△
K
MQ
intersect for the second time at
Y
Y
Y
. The segments
B
P
BP
BP
and
C
Q
CQ
CQ
intersect at point
T
T
T
. Prove that the line
M
T
MT
MT
is tangent to the circumcircle of
△
M
X
Y
\triangle MXY
△
MX
Y
.
1
1
Hide problems
Colour the triangle
A white equilateral triangle
T
T
T
with side length
2022
2022
2022
is divided into equilateral triangles with side
1
1
1
(cells) by lines parallel to the sides of
T
T
T
. We'll call two cells
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
a
d
j
a
c
e
n
t
<
/
s
p
a
n
>
<span class='latex-italic'>adjacent</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
a
d
ja
ce
n
t
<
/
s
p
an
>
if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set
S
S
S
of cells and Ivan tells him the parity of the number of black cells in
S
S
S
. After knowing this, Peter is able to determine the parity of the number of
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
a
d
j
a
c
e
n
t
<
/
s
p
a
n
>
<span class='latex-italic'>adjacent</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
a
d
ja
ce
n
t
<
/
s
p
an
>
cells of different colours. Find all possible cardinalities of
S
S
S
such that this is always possible independent of how Ivan chooses to colour the cells.