MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2007 Bulgaria National Olympiad
2007 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
2
2
Hide problems
The greatest n
Find the greatest positive integer
n
n
n
such that we can choose
2007
2007
2007
different positive integers from
[
2
⋅
1
0
n
−
1
,
1
0
n
)
[2\cdot 10^{n-1},10^{n})
[
2
⋅
1
0
n
−
1
,
1
0
n
)
such that for each two
1
≤
i
<
j
≤
n
1\leq i<j\leq n
1
≤
i
<
j
≤
n
there exists a positive integer
a
1
a
2
…
a
n
‾
\overline{a_{1}a_{2}\ldots a_{n}}
a
1
a
2
…
a
n
from the chosen integers for which
a
j
≥
a
i
+
2
a_{j}\geq a_{i}+2
a
j
≥
a
i
+
2
.A. Ivanov, E. Kolev
Cover equilaterial triangle with 5 equilaterial triangles
Find the least real number
m
m
m
such that with all
5
5
5
equilaterial triangles with sum of areas
m
m
m
we can cover an equilaterial triangle with side 1. O. Mushkarov, N. Nikolov
1
2
Hide problems
Inscribed circle (St. Petersburg, 2002)
The quadrilateral
A
B
C
D
ABCD
A
BC
D
, where
∠
B
A
D
+
∠
A
D
C
>
π
\angle BAD+\angle ADC>\pi
∠
B
A
D
+
∠
A
D
C
>
π
, is inscribed a circle with centre
I
I
I
. A line through
I
I
I
intersects
A
B
AB
A
B
and
C
D
CD
C
D
in points
X
X
X
and
Y
Y
Y
respectively such that
I
X
=
I
Y
IX=IY
I
X
=
I
Y
. Prove that
A
X
⋅
D
Y
=
B
X
⋅
C
Y
AX\cdot DY=BX\cdot CY
A
X
⋅
D
Y
=
BX
⋅
C
Y
.
Good set for all positive integers
Let
k
>
1
k>1
k
>
1
be a given positive integer. A set
S
S
S
of positive integers is called good if we can colour the set of positive integers in
k
k
k
colours such that each integer of
S
S
S
cannot be represented as sum of two positive integers of the same colour. Find the greatest
t
t
t
such that the set
S
=
{
a
+
1
,
a
+
2
,
…
,
a
+
t
}
S=\{a+1,a+2,\ldots ,a+t\}
S
=
{
a
+
1
,
a
+
2
,
…
,
a
+
t
}
is good for all positive integers
a
a
a
.A. Ivanov, E. Kolev
3
2
Hide problems
Trigonometric function not of a special form
Find the least positive integer
n
n
n
such that
cos
π
n
\cos\frac{\pi}{n}
cos
n
π
cannot be written in the form
p
+
q
+
r
3
p+\sqrt{q}+\sqrt[3]{r}
p
+
q
+
3
r
with
p
,
q
,
r
∈
Q
p,q,r\in\mathbb{Q}
p
,
q
,
r
∈
Q
.O. Mushkarov, N. NikolovNo-one in the competition scored more than 2 points
A beautiful one
Let
P
(
x
)
∈
Z
[
x
]
P(x)\in \mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
be a monic polynomial with even degree. Prove that, if for infinitely many integers
x
x
x
, the number
P
(
x
)
P(x)
P
(
x
)
is a square of a positive integer, then there exists a polynomial
Q
(
x
)
∈
Z
[
x
]
Q(x)\in\mathbb{Z}[x]
Q
(
x
)
∈
Z
[
x
]
such that
P
(
x
)
=
Q
(
x
)
2
P(x)=Q(x)^2
P
(
x
)
=
Q
(
x
)
2
.