MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria MO Regional Round
2024 Bulgaria MO Regional Round
2024 Bulgaria MO Regional Round
Part of
Bulgaria MO Regional Round
Subcontests
(10)
12.4
1
Hide problems
Classical NT about density of numbers of a special form
Find all pairs of positive integers
(
n
,
k
)
(n, k)
(
n
,
k
)
such that all sufficiently large odd positive integers
m
m
m
are representable as
m
=
a
1
n
2
+
a
2
(
n
+
1
)
2
+
…
+
a
k
(
n
+
k
−
1
)
2
+
a
k
+
1
(
n
+
k
)
2
m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}
m
=
a
1
n
2
+
a
2
(
n
+
1
)
2
+
…
+
a
k
(
n
+
k
−
1
)
2
+
a
k
+
1
(
n
+
k
)
2
for some non-negative integers
a
1
,
a
2
,
…
,
a
k
+
1
a_1, a_2, \ldots, a_{k+1}
a
1
,
a
2
,
…
,
a
k
+
1
.
12.3
1
Hide problems
Great combination of geometry and analytic arguments
Let
A
0
B
0
C
0
A_0B_0C_0
A
0
B
0
C
0
be a triangle. For a positive integer
n
≥
1
n \geq 1
n
≥
1
, we define
A
n
A_n
A
n
on the segment
B
n
−
1
C
n
−
1
B_{n-1}C_{n-1}
B
n
−
1
C
n
−
1
such that
B
n
−
1
A
n
:
C
n
−
1
A
n
=
2
:
1
B_{n-1}A_n:C_{n-1}A_n=2:1
B
n
−
1
A
n
:
C
n
−
1
A
n
=
2
:
1
and
B
n
,
C
n
B_n, C_n
B
n
,
C
n
are defined cyclically in a similar manner. Show that there exists an unique point
P
P
P
that lies in the interior of all triangles
A
n
B
n
C
n
A_nB_nC_n
A
n
B
n
C
n
.
12.2
1
Hide problems
Nice sequence bound
Let
N
N
N
be a positive integer. The sequence
x
1
,
x
2
,
…
x_1, x_2, \ldots
x
1
,
x
2
,
…
of non-negative reals is defined by
x
n
2
=
∑
i
=
1
n
−
1
x
i
x
n
−
i
x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}
x
n
2
=
i
=
1
∑
n
−
1
x
i
x
n
−
i
for all positive integers
n
>
N
n>N
n
>
N
. Show that there exists a constant
c
>
0
c>0
c
>
0
, such that
x
n
≤
n
2
+
c
x_n \leq \frac{n} {2}+c
x
n
≤
2
n
+
c
for all positive integers
n
n
n
.
12.1
1
Hide problems
Conditional geo with incenters
Let
A
B
C
ABC
A
BC
be an acute triangle with midpoint
M
M
M
of
A
B
AB
A
B
. The point
D
D
D
lies on the segment
M
B
MB
MB
and
I
1
,
I
2
I_1, I_2
I
1
,
I
2
denote the incenters of
△
A
D
C
\triangle ADC
△
A
D
C
and
△
B
D
C
\triangle BDC
△
B
D
C
. Given that
∠
I
1
M
I
2
=
9
0
∘
\angle I_1MI_2=90^{\circ}
∠
I
1
M
I
2
=
9
0
∘
, show that
C
A
=
C
B
CA=CB
C
A
=
CB
.
11.4
1
Hide problems
Classical double-counting combo
A board
2025
×
2025
2025 \times 2025
2025
×
2025
is filled with the numbers
1
,
2
,
…
,
2025
1, 2, \ldots, 2025
1
,
2
,
…
,
2025
, each appearing exactly
2025
2025
2025
times. Show that there is a row or column with at least
45
45
45
distinct numbers.
11.3
1
Hide problems
NT with difference of divisors
A positive integer
n
n
n
is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
if
2
∣
τ
(
n
)
2 \mid \tau(n)
2
∣
τ
(
n
)
and if its divisors are
1
=
d
1
<
d
2
<
…
<
d
2
k
−
1
<
d
2
k
=
n
,
1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n,
1
=
d
1
<
d
2
<
…
<
d
2
k
−
1
<
d
2
k
=
n
,
then
d
k
+
1
−
d
k
=
2
d_{k+1}-d_k=2
d
k
+
1
−
d
k
=
2
and
d
k
+
2
−
d
k
−
1
=
65
d_{k+2}-d_{k-1}=65
d
k
+
2
−
d
k
−
1
=
65
. Find the smallest
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
number.
10.3
1
Hide problems
Special representation of primes and their squares
Find all positive integers
1
≤
k
≤
6
1 \leq k \leq 6
1
≤
k
≤
6
such that for any prime
p
p
p
, satisfying
p
2
=
a
2
+
k
b
2
p^2=a^2+kb^2
p
2
=
a
2
+
k
b
2
for some positive integers
a
,
b
a, b
a
,
b
, there exist positive integers
x
,
y
x, y
x
,
y
, satisfying
p
=
x
2
+
k
y
2
p=x^2+ky^2
p
=
x
2
+
k
y
2
.[hide=Remark on 10.4] It also appears as ARO 2010 10.4 with the grid changed to
10
×
10
10 \times 10
10
×
10
and
17
17
17
changed to
5
5
5
, so it will not be posted.
10.2
1
Hide problems
Locus geo, nicely generalizing the nine-point circle configuration
Given are two fixed lines that meet at a point
O
O
O
and form an acute angle with measure
α
\alpha
α
. Let
P
P
P
be a fixed point, internal for the angle. The points
M
,
N
M, N
M
,
N
vary on the two lines (one point on each line) such that
∠
M
P
N
=
18
0
∘
−
α
\angle MPN=180^{\circ}-\alpha
∠
MPN
=
18
0
∘
−
α
and
P
P
P
is internal for
△
M
O
N
\triangle MON
△
MON
. Show that the foot of the perpendicular from
P
P
P
to
M
N
MN
MN
lies on a fixed circle.
9.4
1
Hide problems
Weighted complete graph
Given is a
K
2024
K_{2024}
K
2024
in which every edge has weight
1
1
1
or
2
2
2
. If every cycle has even total weight, find the minimal value of the sum of all weights in the graph.
9.3
1
Hide problems
Squares consisting of digits 0, 4, 9
A positive integer
n
n
n
is called a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
u
p
e
r
s
q
u
a
r
e
<
/
s
p
a
n
>
<span class='latex-italic'>supersquare</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
u
p
ers
q
u
a
re
<
/
s
p
an
>
if there exists a positive integer
m
m
m
, such that
10
∤
m
10 \nmid m
10
∤
m
and the decimal representation of
n
=
m
2
n=m^2
n
=
m
2
consists only of digits among
{
0
,
4
,
9
}
\{0, 4, 9\}
{
0
,
4
,
9
}
. Are there infinitely many
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
u
p
e
r
s
q
u
a
r
e
s
<
/
s
p
a
n
>
<span class='latex-italic'>supersquares</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
u
p
ers
q
u
a
res
<
/
s
p
an
>
?