MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2011 Bulgaria National Olympiad
2011 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
3
2
Hide problems
Distance inequality holds for all interior points
Triangle
A
B
C
ABC
A
BC
and a function
f
:
R
+
→
R
f:\mathbb{R}^+\to\mathbb{R}
f
:
R
+
→
R
have the following property: for every line segment
D
E
DE
D
E
from the interior of the triangle with midpoint
M
M
M
, the inequality
f
(
d
(
D
)
)
+
f
(
d
(
E
)
)
≤
2
f
(
d
(
M
)
)
f(d(D))+f(d(E))\le 2f(d(M))
f
(
d
(
D
))
+
f
(
d
(
E
))
≤
2
f
(
d
(
M
))
, where
d
(
X
)
d(X)
d
(
X
)
is the distance from point
X
X
X
to the nearest side of the triangle (
X
X
X
is in the interior of
△
A
B
C
\triangle ABC
△
A
BC
). Prove that for each line segment
P
Q
PQ
PQ
and each point interior point
N
N
N
the inequality
∣
Q
N
∣
f
(
d
(
P
)
)
+
∣
P
N
∣
f
(
d
(
Q
)
)
≤
∣
P
Q
∣
f
(
d
(
N
)
)
|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))
∣
QN
∣
f
(
d
(
P
))
+
∣
PN
∣
f
(
d
(
Q
))
≤
∣
PQ
∣
f
(
d
(
N
))
holds.
The number of good colourings in 2011-gon
In the interior of the convex 2011-gon are
2011
2011
2011
points, such that no three among the given
4022
4022
4022
points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied: 1) Each segment joins two points of the same colour. 2) None of the line segments intersect. 3) For any two points of the same colour there exists a path of segments connecting them. Find the number of "good" colourings.
2
2
Hide problems
f_n(x) has 2^n different roots
Let
f
1
(
x
)
f_1(x)
f
1
(
x
)
be a polynomial of degree
2
2
2
with the leading coefficient positive and
f
n
+
1
(
x
)
=
f
1
(
f
n
(
x
)
)
f_{n+1}(x) =f_1(f_n(x))
f
n
+
1
(
x
)
=
f
1
(
f
n
(
x
))
for
n
≥
1.
n\ge 1.
n
≥
1.
Prove that if the equation
f
2
(
x
)
=
0
f_2(x)=0
f
2
(
x
)
=
0
has four different non-positive real roots, then for arbitrary
n
n
n
then
f
n
(
x
)
f_n(x)
f
n
(
x
)
has
2
n
2^n
2
n
different real roots.
τ(φ(n))=φ(τ(n)) where n has exactly two prime divisors
For each natural number
a
a
a
we denote
τ
(
a
)
\tau (a)
τ
(
a
)
and
ϕ
(
a
)
\phi (a)
ϕ
(
a
)
the number of natural numbers dividing
a
a
a
and the number of natural numbers less than
a
a
a
that are relatively prime to
a
a
a
. Find all natural numbers
n
n
n
for which
n
n
n
has exactly two different prime divisors and
n
n
n
satisfies
τ
(
ϕ
(
n
)
)
=
ϕ
(
τ
(
n
)
)
\tau (\phi (n))=\phi (\tau (n))
τ
(
ϕ
(
n
))
=
ϕ
(
τ
(
n
))
.
1
2
Hide problems
Binomial diophantine equation
Prove whether or not there exist natural numbers
n
,
k
n,k
n
,
k
where
1
≤
k
≤
n
−
2
1\le k\le n-2
1
≤
k
≤
n
−
2
such that
(
n
k
)
2
+
(
n
k
+
1
)
2
=
(
n
k
+
2
)
4
\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4
(
k
n
)
2
+
(
k
+
1
n
)
2
=
(
k
+
2
n
)
4
nine-point circle generalized
Point
O
O
O
is inside
△
A
B
C
\triangle ABC
△
A
BC
. The feet of perpendicular from
O
O
O
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
are
D
,
E
,
F
D,E,F
D
,
E
,
F
. Perpendiculars from
A
A
A
and
B
B
B
respectively to
E
F
EF
EF
and
F
D
FD
F
D
meet at
P
P
P
. Let
H
H
H
be the foot of perpendicular from
P
P
P
to
A
B
AB
A
B
. Prove that
D
,
E
,
F
,
H
D,E,F,H
D
,
E
,
F
,
H
are concyclic.