MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Winter Tournament
2024 Bulgarian Winter Tournament
2024 Bulgarian Winter Tournament
Part of
Bulgarian Winter Tournament
Subcontests
(12)
9.1
1
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System of equations with two variables
Find all real
x
,
y
x, y
x
,
y
, satisfying
(
x
+
1
)
2
(
y
+
1
)
2
=
27
x
y
(x+1)^2(y+1)^2=27xy
(
x
+
1
)
2
(
y
+
1
)
2
=
27
x
y
and
(
x
2
+
1
)
(
y
2
+
1
)
=
10
x
y
.
(x^2+1)(y^2+1)=10xy.
(
x
2
+
1
)
(
y
2
+
1
)
=
10
x
y
.
9.2
1
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Maximal ratio of two primes
Let
p
>
q
p>q
p
>
q
be primes, such that
240
∤
p
4
−
q
4
240 \nmid p^4-q^4
240
∤
p
4
−
q
4
. Find the maximal value of
q
p
\frac{q} {p}
p
q
.
9.3
1
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Well-known conditional geo
Let
A
B
C
ABC
A
BC
be a triangle, satisfying
2
A
C
=
A
B
+
B
C
2AC=AB+BC
2
A
C
=
A
B
+
BC
. If
O
O
O
and
I
I
I
are its circumcenter and incenter, show that
∠
O
I
B
=
9
0
∘
\angle OIB=90^{\circ}
∠
O
I
B
=
9
0
∘
.
9.4
1
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Colored segments
There are
11
11
11
points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?
10.2
1
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When this fraction is a prime?
Find all positive integers
k
k
k
for which there exist positive integers
x
,
y
x, y
x
,
y
, such that
x
k
y
x
2
+
y
2
\frac{x^ky}{x^2+y^2}
x
2
+
y
2
x
k
y
is a prime.
10.4
1
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Delete edges to obtain a 2-colorable graph
Let
n
≥
3
n \geq 3
n
≥
3
be a positive integer. Find the smallest positive real
k
k
k
, satisfying the following condition: if
G
G
G
is a connected graph with
n
n
n
vertices and
m
m
m
edges, then it is always possible to delete at most
k
(
m
−
⌊
n
2
⌋
)
k(m-\lfloor \frac{n} {2} \rfloor)
k
(
m
−
⌊
2
n
⌋)
edges, so that the resulting graph has a proper vertex coloring with two colors.
11.3
1
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Can the nominator of this sum be squarefree?
Let
q
>
3
q>3
q
>
3
be a rational number, such that
q
2
−
4
q^2-4
q
2
−
4
is a perfect square of a rational number. The sequence
a
0
,
a
1
,
…
a_0, a_1, \ldots
a
0
,
a
1
,
…
is defined by
a
0
=
2
,
a
1
=
q
,
a
i
+
1
=
q
a
i
−
a
i
−
1
a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}
a
0
=
2
,
a
1
=
q
,
a
i
+
1
=
q
a
i
−
a
i
−
1
for all
i
≥
1
i \geq 1
i
≥
1
. Is it true that there exist a positive integer
n
n
n
and nonzero integers
b
0
,
b
1
,
…
,
b
n
b_0, b_1, \ldots, b_n
b
0
,
b
1
,
…
,
b
n
with sum zero, such that if
∑
i
=
0
n
a
i
b
i
=
A
B
\sum_{i=0}^{n} a_ib_i=\frac{A} {B}
∑
i
=
0
n
a
i
b
i
=
B
A
for
(
A
,
B
)
=
1
(A, B)=1
(
A
,
B
)
=
1
, then
A
A
A
is squarefree?
11.4
1
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Edge-colored Kn has a colorful triangle
Let
n
,
k
n, k
n
,
k
be positive integers with
k
≥
3
k \geq 3
k
≥
3
. The edges of of a complete graph
K
n
K_n
K
n
are colored in
k
k
k
colors, such that for any color
i
i
i
and any two vertices, there exists a path between them, consisting only of edges in color
i
i
i
. Prove that there exist three vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
of
K
n
K_n
K
n
, such that
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
A
CA
C
A
are all distinctly colored.
12.1
1
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Tossing coins
Maria and Bilyana play the following game. Maria has
2024
2024
2024
fair coins and Bilyana has
2023
2023
2023
fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?
12.2
1
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Spiral similarity geo
Let
A
B
C
ABC
A
BC
be scalene and acute triangle with
C
A
>
C
B
CA>CB
C
A
>
CB
and let
P
P
P
be an internal point, satisfying
∠
A
P
B
=
18
0
∘
−
∠
A
C
B
\angle APB=180^{\circ}-\angle ACB
∠
A
PB
=
18
0
∘
−
∠
A
CB
; the lines
A
P
,
B
P
AP, BP
A
P
,
BP
meet
B
C
,
C
A
BC, CA
BC
,
C
A
at
A
1
,
B
1
A_1, B_1
A
1
,
B
1
. If
M
M
M
is the midpoint of
A
1
B
1
A_1B_1
A
1
B
1
and
(
A
1
B
1
C
)
(A_1B_1C)
(
A
1
B
1
C
)
meets
(
A
B
C
)
(ABC)
(
A
BC
)
at
Q
Q
Q
, show that
∠
P
Q
M
=
∠
B
Q
A
1
\angle PQM=\angle BQA_1
∠
PQM
=
∠
BQ
A
1
.
12.3
1
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Probabilistic sum is strictly increasing
Let
n
n
n
be a positive integer and let
A
\mathcal{A}
A
be a family of non-empty subsets of
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots, n \}
{
1
,
2
,
…
,
n
}
such that if
A
∈
A
A \in \mathcal{A}
A
∈
A
and
A
A
A
is subset of a set
B
⊆
{
1
,
2
,
…
,
n
}
B\subseteq \{1, 2, \ldots, n\}
B
⊆
{
1
,
2
,
…
,
n
}
, then
B
B
B
is also in
A
\mathcal{A}
A
. Show that the function
f
(
x
)
:
=
∑
A
∈
A
x
∣
A
∣
(
1
−
x
)
n
−
∣
A
∣
f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|}
f
(
x
)
:=
A
∈
A
∑
x
∣
A
∣
(
1
−
x
)
n
−
∣
A
∣
is strictly increasing for
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
.
12.4
1
Hide problems
np is good for infinitely many primes p
Call a positive integer
m
m
m
<
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 there exist integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
m
=
a
3
+
2
b
3
+
4
c
3
−
6
a
b
c
m=a^3+2b^3+4c^3-6abc
m
=
a
3
+
2
b
3
+
4
c
3
−
6
ab
c
. Show that there exists a positive integer
n
<
2024
n<2024
n
<
2024
, such that for infinitely many primes
p
p
p
, the number
n
p
np
n
p
is
<
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
>
.