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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1993 Bulgaria National Olympiad
1993 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
5
1
Hide problems
functional with geo, f(A_1,A_2) = vector OA_1 x OA_2 wanted
Let
O
x
y
Oxy
O
x
y
be a fixed rectangular coordinate system in the plane. Each ordered pair of points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
from the same plane which are different from O and have coordinates
x
1
,
y
1
x_1, y_1
x
1
,
y
1
and
x
2
,
y
2
x_2, y_2
x
2
,
y
2
respectively is associated with real number
f
(
A
1
,
A
2
)
f(A_1,A_2)
f
(
A
1
,
A
2
)
in such a way that the following conditions are satisfied:(a) If
O
A
1
=
O
B
1
OA_1 = OB_1
O
A
1
=
O
B
1
,
O
A
2
=
O
B
2
OA_2 = OB_2
O
A
2
=
O
B
2
and
A
1
A
2
=
B
1
B
2
A_1A_2 = B_1B_2
A
1
A
2
=
B
1
B
2
then
f
(
A
1
,
A
2
)
=
f
(
B
1
,
B
2
)
f(A_1,A_2) = f(B_1,B_2)
f
(
A
1
,
A
2
)
=
f
(
B
1
,
B
2
)
.(b) There exists a polynomial of second degree
F
(
u
,
v
,
w
,
z
)
F(u,v,w,z)
F
(
u
,
v
,
w
,
z
)
such that
f
(
A
1
,
A
2
)
=
F
(
x
1
,
y
1
,
x
2
,
y
2
)
f(A_1,A_2)=F(x_1,y_1,x_2,y_2)
f
(
A
1
,
A
2
)
=
F
(
x
1
,
y
1
,
x
2
,
y
2
)
.(c) There exists such a number
ϕ
∈
(
0
,
π
)
\phi \in (0,\pi)
ϕ
∈
(
0
,
π
)
that for every two points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
for which
∠
A
1
O
A
2
=
ϕ
\angle A_1OA_2 = \phi
∠
A
1
O
A
2
=
ϕ
is satisfied
f
(
A
1
,
A
2
)
=
0
f(A_1,A_2) = 0
f
(
A
1
,
A
2
)
=
0
.(d) If the points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
are such that the triangle
O
A
1
A
2
OA_1A_2
O
A
1
A
2
is equilateral with side
1
1
1
then
f
(
A
1
,
A
2
)
=
1
2
f(A_1,A_2) = \frac12
f
(
A
1
,
A
2
)
=
2
1
.Prove that
f
(
A
1
,
A
2
)
=
O
A
1
→
⋅
O
A
2
→
f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}
f
(
A
1
,
A
2
)
=
O
A
1
⋅
O
A
2
for each ordered pair of points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
.
6
1
Hide problems
for every A, exist 3 points X,Y,Z such that AX = AY = AZ = 1
Find all natural numbers
n
n
n
for which there exists set
S
S
S
consisting of
n
n
n
points in the plane, satisfying the condition: For each point
A
∈
S
A \in S
A
∈
S
there exist at least three points say
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
from
S
S
S
such that the segments
A
X
,
A
Y
AX, AY
A
X
,
A
Y
and
A
Z
AZ
A
Z
have length
1
1
1
(it means that
A
X
=
A
Y
=
A
Z
=
1
AX = AY = AZ = 1
A
X
=
A
Y
=
A
Z
=
1
).
4
1
Hide problems
a_n such that {a_i +a_j} form a full system modulo n(n+1)/2
Find all natural numbers
n
>
1
n > 1
n
>
1
for which there exists such natural numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
for which the numbers
{
a
i
+
a
j
∣
1
≤
i
≤
j
≤
n
}
\{a_i +a_j | 1 \le i \le j \le n \}
{
a
i
+
a
j
∣1
≤
i
≤
j
≤
n
}
form a full system modulo
n
(
n
+
1
)
2
\frac{n(n+1)}{2}
2
n
(
n
+
1
)
.
3
1
Hide problems
coloring into 2 colours a polyhedral from 2 regular pyramids with bases 7-gon
it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with
7
7
7
vertices) with common base
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A_1A_2A_3A_4A_5A_6A_7
A
1
A
2
A
3
A
4
A
5
A
6
A
7
and vertices respectively the points
B
B
B
and
C
C
C
. The edges
B
A
i
,
C
A
i
BA_i , CA_i
B
A
i
,
C
A
i
(
i
=
1
,
.
.
.
,
7
(i = 1,...,7
(
i
=
1
,
...
,
7
), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.
1
1
Hide problems
f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n)), f(1)=1
Find all functions
f
f
f
, defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
; (b) for every two whole (integer) numbers
m
m
m
and
n
n
n
, the following equality is satisfied:
f
(
m
+
n
)
⋅
(
f
(
m
)
−
f
(
n
)
)
=
f
(
m
−
n
)
⋅
(
f
(
m
)
+
f
(
n
)
)
f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))
f
(
m
+
n
)
⋅
(
f
(
m
)
−
f
(
n
))
=
f
(
m
−
n
)
⋅
(
f
(
m
)
+
f
(
n
))
2
1
Hide problems
Geometric inequality proof
Let
M
M
M
be an interior point of the triangle
A
B
C
ABC
A
BC
such that
A
M
C
=
9
0
∘
AMC = 90^\circ
A
MC
=
9
0
∘
,
A
M
B
=
15
0
∘
AMB = 150^\circ
A
MB
=
15
0
∘
, and
B
M
C
=
12
0
∘
BMC = 120^\circ
BMC
=
12
0
∘
. The circumcenters of the triangles
A
M
C
AMC
A
MC
,
A
M
B
AMB
A
MB
, and
B
M
C
BMC
BMC
are
P
P
P
,
Q
Q
Q
, and
R
R
R
respectively. Prove that the area of
Δ
P
Q
R
\Delta PQR
Δ
PQR
is greater than or equal to the area of
Δ
A
B
C
\Delta ABC
Δ
A
BC
.