MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2020 Bulgaria Team Selection Test
2020 Bulgaria Team Selection Test
Part of
Bulgaria Team Selection Test
Subcontests
(5)
6
1
Hide problems
Excenters and concyclic
In triangle
△
A
B
C
\triangle ABC
△
A
BC
,
B
C
>
A
C
BC>AC
BC
>
A
C
,
I
B
I_B
I
B
is the
B
B
B
-excenter, the line through
C
C
C
parallel to
A
B
AB
A
B
meets
B
I
B
BI_B
B
I
B
at
F
F
F
.
M
M
M
is the midpoint of
A
I
B
AI_B
A
I
B
and the
A
A
A
-excircle touches side
A
B
AB
A
B
at
D
D
D
. Point
E
E
E
satisfies
∠
B
A
C
=
∠
B
D
E
,
D
E
=
B
C
\angle BAC=\angle BDE, DE=BC
∠
B
A
C
=
∠
B
D
E
,
D
E
=
BC
, and lies on the same side as
C
C
C
of
A
B
AB
A
B
. Let
E
C
EC
EC
intersect
A
B
,
F
M
AB,FM
A
B
,
FM
at
P
,
Q
P,Q
P
,
Q
respectively. Prove that
P
,
A
,
M
,
Q
P,A,M,Q
P
,
A
,
M
,
Q
are concyclic.
1
1
Hide problems
Equal angles iff ratio of lengths are equal
In acute triangle
△
A
B
C
\triangle ABC
△
A
BC
,
B
C
>
A
C
BC>AC
BC
>
A
C
,
Γ
\Gamma
Γ
is its circumcircle,
D
D
D
is a point on segment
A
C
AC
A
C
and
E
E
E
is the intersection of the circle with diameter
C
D
CD
C
D
and
Γ
\Gamma
Γ
.
M
M
M
is the midpoint of
A
B
AB
A
B
and
C
M
CM
CM
meets
Γ
\Gamma
Γ
again at
Q
Q
Q
. The tangents to
Γ
\Gamma
Γ
at
A
,
B
A,B
A
,
B
meet at
P
P
P
, and
H
H
H
is the foot of perpendicular from
P
P
P
to
B
Q
BQ
BQ
.
K
K
K
is a point on line
H
Q
HQ
H
Q
such that
Q
Q
Q
lies between
H
H
H
and
K
K
K
. Prove that
∠
H
K
P
=
∠
A
C
E
\angle HKP=\angle ACE
∠
HK
P
=
∠
A
CE
if and only if
K
Q
Q
H
=
C
D
D
A
\frac{KQ}{QH}=\frac{CD}{DA}
Q
H
K
Q
=
D
A
C
D
.
3
1
Hide problems
A family of sets
Let
C
\mathcal{C}
C
be a family of subsets of
A
=
{
1
,
2
,
…
,
100
}
A=\{1,2,\dots,100\}
A
=
{
1
,
2
,
…
,
100
}
satisfying the following two conditions:1) Every
99
99
99
element subset of
A
A
A
is in
C
.
\mathcal{C}.
C
.
2) For any non empty subset
C
∈
C
C\in\mathcal{C}
C
∈
C
there is
c
∈
C
c\in C
c
∈
C
such that
C
∖
{
c
}
∈
C
.
C\setminus\{c\}\in \mathcal{C}.
C
∖
{
c
}
∈
C
.
What is the least possible value of
∣
C
∣
|\mathcal{C}|
∣
C
∣
?
2
1
Hide problems
Multiple of power of two.
Given two odd natural numbers
a
,
b
a,b
a
,
b
prove that for each
n
∈
N
n\in\mathbb{N}
n
∈
N
there exists
m
∈
N
m\in\mathbb{N}
m
∈
N
such that either
a
m
b
2
−
1
a^mb^2-1
a
m
b
2
−
1
or
b
m
a
2
−
1
b^ma^2-1
b
m
a
2
−
1
is multiple of
2
n
.
2^n.
2
n
.
5
1
Hide problems
|f(x+y)-f(x)-f(y)|<=1, approximate by an additive function
Given is a function
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
≤
1
|f(x+y)-f(x)-f(y)|\leq 1
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
≤
1
. Prove the existence of an additive function
g
:
R
→
R
g:\mathbb{R}\rightarrow \mathbb{R}
g
:
R
→
R
(that is
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
g(x+y)=g(x)+g(y)
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
) such that
∣
f
(
x
)
−
g
(
x
)
∣
≤
1
|f(x)-g(x)|\leq 1
∣
f
(
x
)
−
g
(
x
)
∣
≤
1
for any
x
∈
R
x \in \mathbb{R}
x
∈
R