MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2023 Bulgaria National Olympiad
2023 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
6
1
Hide problems
Contrived C strikes again
In a class of
26
26
26
students, everyone is being graded on five subjects with one of three possible marks. Prove that if
25
25
25
of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.
5
1
Hide problems
n-variable modular inequality
For every positive integer
n
n
n
determine the least possible value of the expression
∣
x
1
∣
+
∣
x
1
−
x
2
∣
+
∣
x
1
+
x
2
−
x
3
∣
+
⋯
+
∣
x
1
+
x
2
+
⋯
+
x
n
−
1
−
x
n
∣
|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|
∣
x
1
∣
+
∣
x
1
−
x
2
∣
+
∣
x
1
+
x
2
−
x
3
∣
+
⋯
+
∣
x
1
+
x
2
+
⋯
+
x
n
−
1
−
x
n
∣
given that
x
1
,
x
2
,
…
,
x
n
x_{1}, x_{2}, \dots , x_{n}
x
1
,
x
2
,
…
,
x
n
are real numbers satisfying
∣
x
1
∣
+
∣
x
2
∣
+
⋯
+
∣
x
n
∣
=
1
|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1
∣
x
1
∣
+
∣
x
2
∣
+
⋯
+
∣
x
n
∣
=
1
.
4
1
Hide problems
Bash Bash Bash
Prove that there exists a unique point
M
M
M
on the side
A
D
AD
A
D
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
S
A
B
M
+
S
C
D
M
=
S
A
B
C
D
\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}
S
A
BM
+
S
C
D
M
=
S
A
BC
D
if and only if
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
.
3
1
Hide problems
Classical problem on integer polynomial
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with positive integer coefficients. For every
n
∈
N
n\in\mathbb{N}
n
∈
N
, let
a
1
(
n
)
,
a
2
(
n
)
,
…
,
a
n
(
n
)
a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}
a
1
(
n
)
,
a
2
(
n
)
,
…
,
a
n
(
n
)
be fixed positive integers that give pairwise different residues modulo
n
n
n
and let
g
(
n
)
=
∑
i
=
1
n
f
(
a
i
(
n
)
)
=
f
(
a
1
(
n
)
)
+
f
(
a
2
(
n
)
)
+
⋯
+
f
(
a
n
(
n
)
)
g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})
g
(
n
)
=
i
=
1
∑
n
f
(
a
i
(
n
)
)
=
f
(
a
1
(
n
)
)
+
f
(
a
2
(
n
)
)
+
⋯
+
f
(
a
n
(
n
)
)
Prove that there exists a constant
M
M
M
such that for all integers
m
>
M
m>M
m
>
M
we have
gcd
(
m
,
g
(
m
)
)
>
202
3
2023
\gcd(m, g(m))>2023^{2023}
g
cd
(
m
,
g
(
m
))
>
202
3
2023
.
2
1
Hide problems
Excircles Config Geo
Let
A
B
C
ABC
A
BC
be an acute triangle and
A
1
,
B
1
,
C
1
A_{1}, B_{1}, C_{1}
A
1
,
B
1
,
C
1
be the touchpoints of the excircles with the segments
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Let
O
A
,
O
B
,
O
C
O_{A}, O_{B}, O_{C}
O
A
,
O
B
,
O
C
be the circumcenters of
△
A
B
1
C
1
,
△
B
C
1
A
1
,
△
C
A
1
B
1
\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}
△
A
B
1
C
1
,
△
B
C
1
A
1
,
△
C
A
1
B
1
respectively. Prove that the lines through
O
A
,
O
B
,
O
C
O_{A}, O_{B}, O_{C}
O
A
,
O
B
,
O
C
respectively parallel to the internal angle bisectors of
∠
A
,
∠
B
,
∠
C
\angle A,\angle B, \angle C
∠
A
,
∠
B
,
∠
C
are concurrent.
1
1
Hide problems
Gcd of cycle lengths
Let
G
G
G
be a graph on
n
≥
6
n\geq 6
n
≥
6
vertices and every vertex is of degree at least 3. If
C
1
,
C
2
,
…
,
C
k
C_{1}, C_{2}, \dots, C_{k}
C
1
,
C
2
,
…
,
C
k
are all the cycles in
G
G
G
, determine all possible values of
gcd
(
∣
C
1
∣
,
∣
C
2
∣
,
…
,
∣
C
k
∣
)
\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)
g
cd
(
∣
C
1
∣
,
∣
C
2
∣
,
…
,
∣
C
k
∣
)
where
∣
C
∣
|C|
∣
C
∣
denotes the number of vertices in the cycle
C
C
C
.