MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2007 Bulgaria Team Selection Test
2007 Bulgaria Team Selection Test
Part of
Bulgaria Team Selection Test
Subcontests
(4)
4
2
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Graph
Let
G
G
G
is a graph and
x
x
x
is a vertex of
G
G
G
. Define the transformation
φ
x
\varphi_{x}
φ
x
over
G
G
G
as deleting all incident edges with respect of
x
x
x
and drawing the edges
x
y
xy
x
y
such that
y
∈
G
y\in G
y
∈
G
and
y
y
y
is not connected with
x
x
x
with edge in the beginning of the transformation. A graph
H
H
H
is called
G
−
G-
G
−
attainable if there exists a sequece of such transformations which transforms
G
G
G
in
H
.
H.
H
.
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
and
4
∣
n
.
4|n.
4∣
n
.
Prove that for each graph
G
G
G
with
4
n
4n
4
n
vertices and
n
n
n
edges there exists
G
−
G-
G
−
attainable graph with at least
9
n
2
/
4
9n^{2}/4
9
n
2
/4
triangles.
Quadratic residues
Let
p
=
4
k
+
3
p=4k+3
p
=
4
k
+
3
be a prime number. Find the number of different residues mod p of
(
x
2
+
y
2
)
2
(x^{2}+y^{2})^{2}
(
x
2
+
y
2
)
2
where
(
x
,
p
)
=
(
y
,
p
)
=
1.
(x,p)=(y,p)=1.
(
x
,
p
)
=
(
y
,
p
)
=
1.
3
2
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Nagel Point
Let
I
I
I
be the center of the incircle of non-isosceles triangle
A
B
C
,
A
1
=
A
I
∩
B
C
ABC,A_{1}=AI\cap BC
A
BC
,
A
1
=
A
I
∩
BC
and
B
1
=
B
I
∩
A
C
.
B_{1}=BI\cap AC.
B
1
=
B
I
∩
A
C
.
Let
l
a
l_{a}
l
a
be the line through
A
1
A_{1}
A
1
which is parallel to
A
C
AC
A
C
and
l
b
l_{b}
l
b
be the line through
B
1
B_{1}
B
1
parallel to
B
C
.
BC.
BC
.
Let
l
a
∩
C
I
=
A
2
l_{a}\cap CI=A_{2}
l
a
∩
C
I
=
A
2
and
l
b
∩
C
I
=
B
2
.
l_{b}\cap CI=B_{2}.
l
b
∩
C
I
=
B
2
.
Also
N
=
A
A
2
∩
B
B
2
N=AA_{2}\cap BB_{2}
N
=
A
A
2
∩
B
B
2
and
M
M
M
is the midpoint of
A
B
.
AB.
A
B
.
If
C
N
∥
I
M
CN\parallel IM
CN
∥
I
M
find
C
N
I
M
\frac{CN}{IM}
I
M
CN
.
Easy one
Let
n
≥
2
n\geq 2
n
≥
2
is positive integer. Find the best constant
C
(
n
)
C(n)
C
(
n
)
such that
∑
i
=
1
n
x
i
≥
C
(
n
)
∑
1
≤
j
<
i
≤
n
(
2
x
i
x
j
+
x
i
x
j
)
\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})
i
=
1
∑
n
x
i
≥
C
(
n
)
1
≤
j
<
i
≤
n
∑
(
2
x
i
x
j
+
x
i
x
j
)
is true for all real numbers
x
i
∈
(
0
,
1
)
,
i
=
1
,
.
.
.
,
n
x_{i}\in(0,1),i=1,...,n
x
i
∈
(
0
,
1
)
,
i
=
1
,
...
,
n
for which
(
1
−
x
i
)
(
1
−
x
j
)
≥
1
4
,
1
≤
j
<
i
≤
n
.
(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.
(
1
−
x
i
)
(
1
−
x
j
)
≥
4
1
,
1
≤
j
<
i
≤
n
.
2
2
Hide problems
Functional Inequality
Find all
a
∈
R
a\in\mathbb{R}
a
∈
R
for which there exists a non-constant function
f
:
(
0
,
1
]
→
R
f: (0,1]\rightarrow\mathbb{R}
f
:
(
0
,
1
]
→
R
such that
a
+
f
(
x
+
y
−
x
y
)
+
f
(
x
)
f
(
y
)
≤
f
(
x
)
+
f
(
y
)
a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)
a
+
f
(
x
+
y
−
x
y
)
+
f
(
x
)
f
(
y
)
≤
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
(
0
,
1
]
.
x,y\in(0,1].
x
,
y
∈
(
0
,
1
]
.
k-element subsets = (2n-k)*(2-element)
Let
n
,
k
n,k
n
,
k
be positive integers such that
n
≥
2
k
>
3
n\geq2k>3
n
≥
2
k
>
3
and
A
=
{
1
,
2
,
.
.
.
,
n
}
.
A= \{1,2,...,n\}.
A
=
{
1
,
2
,
...
,
n
}
.
Find all
n
n
n
and
k
k
k
such that the number of
k
k
k
-element subsets of
A
A
A
is
2
n
−
k
2n-k
2
n
−
k
times bigger than the number of
2
2
2
-element subsets of
A
.
A.
A
.
1
2
Hide problems
Extremal geometry problem
Let
A
B
C
ABC
A
BC
is a triangle with
∠
B
A
C
=
π
6
\angle BAC=\frac{\pi}{6}
∠
B
A
C
=
6
π
and the circumradius equal to 1. If
X
X
X
is a point inside or in its boundary let
m
(
X
)
=
min
(
A
X
,
B
X
,
C
X
)
.
m(X)=\min(AX,BX,CX).
m
(
X
)
=
min
(
A
X
,
BX
,
CX
)
.
Find all the angles of this triangle if
max
(
m
(
X
)
)
=
3
3
.
\max(m(X))=\frac{\sqrt{3}}{3}.
max
(
m
(
X
))
=
3
3
.
Isosceles triangle
In isosceles triangle
A
B
C
(
A
C
=
B
C
)
ABC(AC=BC)
A
BC
(
A
C
=
BC
)
the point
M
M
M
is in the segment
A
B
AB
A
B
such that
A
M
=
2
M
B
,
AM=2MB,
A
M
=
2
MB
,
F
F
F
is the midpoint of
B
C
BC
BC
and
H
H
H
is the orthogonal projection of
M
M
M
in
A
F
.
AF.
A
F
.
Prove that
∠
B
H
F
=
∠
A
B
C
.
\angle BHF=\angle ABC.
∠
B
H
F
=
∠
A
BC
.