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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2020 Bulgaria National Olympiad
2020 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
P6
1
Hide problems
Infinite digit number having every n digit number
Let
f
(
x
)
f(x)
f
(
x
)
be a nonconstant real polynomial. The sequence
{
a
i
}
i
=
1
∞
\{a_i\}_{i=1}^{\infty}
{
a
i
}
i
=
1
∞
of real numbers is strictly increasing and unbounded, as
a
i
+
1
<
a
i
+
2020.
a_{i+1}<a_i+2020.
a
i
+
1
<
a
i
+
2020.
The integers
⌊
∣
f
(
a
1
)
∣
⌋
\lfloor{|f(a_1)|}\rfloor
⌊
∣
f
(
a
1
)
∣
⌋
,
⌊
∣
f
(
a
2
)
∣
⌋
\lfloor{|f(a_2)|}\rfloor
⌊
∣
f
(
a
2
)
∣
⌋
,
⌊
∣
f
(
a
3
)
∣
⌋
\lfloor{|f(a_3)|}\rfloor
⌊
∣
f
(
a
3
)
∣
⌋
,
…
\dots
…
are written consecutively in such a way that their digits form an infinite sequence of digits
{
s
k
}
k
=
1
∞
\{s_k\}_{k=1}^{\infty}
{
s
k
}
k
=
1
∞
(here
s
k
∈
{
0
,
1
,
…
,
9
}
s_k\in\{0, 1, \dots, 9\}
s
k
∈
{
0
,
1
,
…
,
9
}
). If
n
∈
N
n\in\mathbb{N}
n
∈
N
, prove that among the numbers
s
n
(
k
−
1
)
+
1
s
n
(
k
−
1
)
+
2
⋯
s
n
k
‾
\overline{s_{n(k-1)+1}s_{n(k-1)+2}\cdots s_{nk}}
s
n
(
k
−
1
)
+
1
s
n
(
k
−
1
)
+
2
⋯
s
nk
, where
k
∈
N
k\in\mathbb{N}
k
∈
N
, all
n
n
n
-digit numbers appear.
P5
1
Hide problems
Changing 2 Hamiltonian cycles
There are
n
n
n
points in the plane, some of which are connected by segments. Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the
n
n
n
points exactly once. It is known that the segments
A
B
AB
A
B
and
B
C
BC
BC
are white. Prove that it is possible to recolor the segments in red and blue in such a way that
A
B
AB
A
B
and
B
C
BC
BC
are recolored as red, [hide=not all of which segments are recolored red]meaning that recoloring every white as red and every black as blue is not acceptable, and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the
n
n
n
points exactly once.
P4
1
Hide problems
Binomial coefficients and squares
Are there positive integers
m
>
4
m>4
m
>
4
and
n
n
n
, such that a)
(
m
3
)
=
n
2
;
{m \choose 3}=n^2;
(
3
m
)
=
n
2
;
b)
(
m
4
)
=
n
2
+
9
?
{m \choose 4}=n^2+9?
(
4
m
)
=
n
2
+
9
?
P2
1
Hide problems
Inequality for real number and integers
Let
b
1
b_1
b
1
,
…
\dots
…
,
b
n
b_n
b
n
be nonnegative integers with sum
2
2
2
and
a
0
a_0
a
0
,
a
1
a_1
a
1
,
…
\dots
…
,
a
n
a_n
a
n
be real numbers such that
a
0
=
a
n
=
0
a_0=a_n=0
a
0
=
a
n
=
0
and
∣
a
i
−
a
i
−
1
∣
≤
b
i
|a_i-a_{i-1}|\leq b_i
∣
a
i
−
a
i
−
1
∣
≤
b
i
for each
i
=
1
i=1
i
=
1
,
…
\dots
…
,
n
n
n
. Prove that
∑
i
=
1
n
(
a
i
+
a
i
−
1
)
b
i
≤
2
\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2
i
=
1
∑
n
(
a
i
+
a
i
−
1
)
b
i
≤
2
I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold
P3
1
Hide problems
Integer sequence and Coprimity
Let
a
1
∈
Z
a_1\in\mathbb{Z}
a
1
∈
Z
,
a
2
=
a
1
2
−
a
1
−
1
a_2=a_1^2-a_1-1
a
2
=
a
1
2
−
a
1
−
1
,
…
\dots
…
,
a
n
+
1
=
a
n
2
−
a
n
−
1
a_{n+1}=a_n^2-a_n-1
a
n
+
1
=
a
n
2
−
a
n
−
1
. Prove that
a
n
+
1
a_{n+1}
a
n
+
1
and
2
n
+
1
2n+1
2
n
+
1
are coprime.
P1
1
Hide problems
Elementary Geometry
On the sides of
△
A
B
C
\triangle{ABC}
△
A
BC
points
P
,
Q
∈
A
B
P,Q \in{AB}
P
,
Q
∈
A
B
(
P
P
P
is between
A
A
A
and
Q
Q
Q
) and
R
∈
B
C
R\in{BC}
R
∈
BC
are chosen. The points
M
M
M
and
N
N
N
are defined as the intersection point of
A
R
AR
A
R
with the segments
C
P
CP
CP
and
C
Q
CQ
CQ
, respectively. If
B
C
=
B
Q
BC=BQ
BC
=
BQ
,
C
P
=
A
P
CP=AP
CP
=
A
P
,
C
R
=
C
N
CR=CN
CR
=
CN
and
∠
B
P
C
=
∠
C
R
A
\angle{BPC}=\angle{CRA}
∠
BPC
=
∠
CR
A
, prove that
M
P
+
N
Q
=
B
R
MP+NQ=BR
MP
+
NQ
=
BR
.