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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Spring Mathematical Competition
2024 Bulgarian Spring Mathematical Competition
2024 Bulgarian Spring Mathematical Competition
Part of
Bulgarian Spring Mathematical Competition
Subcontests
(9)
11.4
1
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Two numbers sum to 0 in some triangle
Given is a convex
2024
2024
2024
-gon
A
1
A
2
…
A
2024
A_1A_2\ldots A_{2024}
A
1
A
2
…
A
2024
and
1000
1000
1000
points inside it, so that no three points are collinear. Some pairs of the points are connected with segments so that the interior of the polygon is divided into triangles. Every point is assigned one number among
{
1
,
−
1
,
2
,
−
2
}
\{1, -1, 2, - 2\}
{
1
,
−
1
,
2
,
−
2
}
, so that the sum of the numbers written in
A
i
A_i
A
i
and
A
i
+
1012
A_{i+1012}
A
i
+
1012
is zero for all
i
=
1
,
2
,
…
,
1012
i=1,2, \ldots, 1012
i
=
1
,
2
,
…
,
1012
. Prove that there is a triangle, such that the sum of the numbers in some two of its vertices is zero.[hide=Remark on source of 11.3] It appears as Estonia TST 2004/5, so it will not be posted.
11.2
1
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Conditional geometry with parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and a circle
k
k
k
passes through
A
,
C
A, C
A
,
C
and meets rays
A
B
,
A
D
AB, AD
A
B
,
A
D
at
E
,
F
E, F
E
,
F
. If
B
D
,
E
F
BD, EF
B
D
,
EF
and the tangent at
C
C
C
concur, show that
A
C
AC
A
C
is diameter of
k
k
k
.
10.4
1
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Divisibility graph
A graph
G
G
G
is called
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=
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<
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>
<span class='latex-italic'>divisibility graph</span>
<
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=
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−
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>
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>
if the vertices can be assigned distinct positive integers such that between two vertices assigned
u
,
v
u, v
u
,
v
there is an edge iff
u
v
\frac{u} {v}
v
u
or
v
u
\frac{v} {u}
u
v
is a positive integer. Show that for any positive integer
n
n
n
and
0
≤
e
≤
n
(
n
−
1
)
2
0 \leq e \leq \frac{n(n-1)}{2}
0
≤
e
≤
2
n
(
n
−
1
)
, there is a
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p
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a
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s
=
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<span class='latex-italic'>divisibility graph</span>
<
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=
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x
−
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>
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<
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>
with
n
n
n
vertices and
e
e
e
edges.[hide=Remark on source of 10.3] It appears to be Kvant 2022 Issue 10 M2719, so it will not be posted; the same problem was also used as 9.4.
10.2
1
Hide problems
Varying circle through a vertex and the incenter and a fixed angle
Let
A
B
C
ABC
A
BC
be a triangle and a circle
ω
\omega
ω
through
C
C
C
and its incenter
I
I
I
meets
C
A
,
C
B
CA, CB
C
A
,
CB
at
P
,
Q
P, Q
P
,
Q
. The circumcircles
(
C
P
Q
)
(CPQ)
(
CPQ
)
and
(
A
B
C
)
(ABC)
(
A
BC
)
meet at
L
L
L
. The angle bisector of
∠
A
L
B
\angle ALB
∠
A
L
B
meets
A
B
AB
A
B
at
K
K
K
. Show that, as
ω
\omega
ω
varies,
∠
P
K
Q
\angle PKQ
∠
P
K
Q
is constant.
10.1
1
Hide problems
Maximal and minimal value of x+2y
The reals
x
,
y
x, y
x
,
y
satisfy
x
(
x
−
6
)
≤
y
(
4
−
y
)
+
7
x(x-6)\leq y(4-y)+7
x
(
x
−
6
)
≤
y
(
4
−
y
)
+
7
. Find the minimal and maximal values of the expression
x
+
2
y
x+2y
x
+
2
y
.
12.4
1
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Graph of binary strings
Let
d
≥
3
d \geq 3
d
≥
3
be a positive integer. The binary strings of length
d
d
d
are splitted into
2
d
−
1
2^{d-1}
2
d
−
1
pairs, such that the strings in each pair differ in exactly one position. Show that there exists an
<
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a
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<span class='latex-italic'>alternating cycle</span>
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=
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−
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>
a
lt
er
na
t
in
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cyc
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<
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>
of length at most
2
d
−
2
2d-2
2
d
−
2
, i.e. at most
2
d
−
2
2d-2
2
d
−
2
binary strings that can be arranged on a circle so that any pair of adjacent strings differ in exactly one position and exactly half of the pairs of adjacent strings are pairs in the split.
12.3
1
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Representation as sums of 33-rd powers
For a positive integer
n
n
n
, denote with
b
(
n
)
b(n)
b
(
n
)
the smallest positive integer
k
k
k
, such that there exist integers
a
1
,
a
2
,
…
,
a
k
a_1, a_2, \ldots, a_k
a
1
,
a
2
,
…
,
a
k
, satisfying
n
=
a
1
33
+
a
2
33
+
…
+
a
k
33
n=a_1^{33}+a_2^{33}+\ldots+a_k^{33}
n
=
a
1
33
+
a
2
33
+
…
+
a
k
33
. Determine whether the set of positive integers
n
n
n
is finite or infinite, which satisfy:a)
b
(
n
)
=
12
;
b(n)=12;
b
(
n
)
=
12
;
b)
b
(
n
)
=
1
2
1
2
12
.
b(n)=12^{12^{12}}.
b
(
n
)
=
1
2
1
2
12
.
12.2
1
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Iff angle conditions geometry
Given is a triangle
A
B
C
ABC
A
BC
and two points
D
∈
A
C
,
E
∈
B
D
D \in AC, E \in BD
D
∈
A
C
,
E
∈
B
D
such that
∠
D
A
E
=
∠
A
E
D
=
∠
A
B
C
\angle DAE=\angle AED=\angle ABC
∠
D
A
E
=
∠
A
E
D
=
∠
A
BC
. Show that
B
E
=
2
C
D
BE=2CD
BE
=
2
C
D
iff
∠
A
C
B
=
9
0
∘
\angle ACB=90^{\circ}
∠
A
CB
=
9
0
∘
.
12.1
1
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Integer terms of a sequence
Given is a sequence
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
, such that
a
1
=
1
a_1=1
a
1
=
1
and
a
n
+
1
=
9
a
n
+
4
a
n
+
6
a_{n+1}=\frac{9a_n+4}{a_n+6}
a
n
+
1
=
a
n
+
6
9
a
n
+
4
for any
n
∈
N
n \in \mathbb{N}
n
∈
N
. Which terms of this sequence are positive integers?