MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1998 Bulgaria National Olympiad
1998 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
3
2
Hide problems
Regular polygons!
On the sides of a non-obtuse triangle
A
B
C
ABC
A
BC
a square, a regular
n
n
n
-gon and a regular
m
m
m
-gon (
m
m
m
,
n
>
5
n > 5
n
>
5
) are constructed externally, so that their centers are vertices of a regular triangle. Prove that
m
=
n
=
6
m = n = 6
m
=
n
=
6
and find the angles of
△
A
B
C
\triangle ABC
△
A
BC
.
Regular polygon coloring!
The sides and diagonals of a regular
n
n
n
-gon
R
R
R
are colored in
k
k
k
colors so that: (i) For each color
a
a
a
and any two vertices
A
A
A
,
B
B
B
of
R
R
R
, the segment
A
B
AB
A
B
is of color
a
a
a
or there is a vertex
C
C
C
such that
A
C
AC
A
C
and
B
C
BC
BC
are of color
a
a
a
. (ii) The sides of any triangle with vertices at vertices of
R
R
R
are colored in at most two colors. Prove that
k
≤
2
k\leq 2
k
≤
2
.
1
2
Hide problems
Binary sequences!
Let
n
n
n
be a natural number. Find the least natural number
k
k
k
for which there exist
k
k
k
sequences of
0
0
0
and
1
1
1
of length
2
n
+
2
2n+2
2
n
+
2
with the following property: any sequence of
0
0
0
and
1
1
1
of length
2
n
+
2
2n+2
2
n
+
2
coincides with some of these
k
k
k
sequences in at least
n
+
2
n+2
n
+
2
positions.
At most one nonzero real root!
Let
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
be real numbers, not all zero. Prove that the equation:
1
+
a
1
x
+
1
+
a
2
x
+
⋯
+
1
+
a
n
x
=
n
\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n
1
+
a
1
x
+
1
+
a
2
x
+
⋯
+
1
+
a
n
x
=
n
has at most one real nonzero root.
2
2
Hide problems
Prove that P(x,y)=P(y,x)
The polynomials
P
n
(
x
,
y
)
,
n
=
1
,
2
,
.
.
.
P_n(x,y), n=1,2,...
P
n
(
x
,
y
)
,
n
=
1
,
2
,
...
are defined by
P
1
(
x
,
y
)
=
1
,
P
n
+
1
(
x
,
y
)
=
(
x
+
y
−
1
)
(
y
+
1
)
P
n
(
x
,
y
+
2
)
+
(
y
−
y
2
)
P
n
(
x
,
y
)
P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)
P
1
(
x
,
y
)
=
1
,
P
n
+
1
(
x
,
y
)
=
(
x
+
y
−
1
)
(
y
+
1
)
P
n
(
x
,
y
+
2
)
+
(
y
−
y
2
)
P
n
(
x
,
y
)
Prove that
P
n
(
x
,
y
)
=
P
n
(
y
,
x
)
P_{n}(x,y)=P_{n}(y,x)
P
n
(
x
,
y
)
=
P
n
(
y
,
x
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
and
n
n
n
.
A is odd
let m and n be natural numbers such that:
3
m
∣
(
m
+
3
)
n
+
1
3m|(m+3)^n+1
3
m
∣
(
m
+
3
)
n
+
1
Prove that
(
m
+
3
)
n
+
1
3
m
\frac{(m+3)^n+1}{3m}
3
m
(
m
+
3
)
n
+
1
is odd