MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2017 Bulgaria National Olympiad
2017 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
6
1
Hide problems
a triangle with altitudes
An acute non-isosceles
△
A
B
C
\triangle ABC
△
A
BC
is given.
C
D
,
A
E
,
B
F
CD, AE, BF
C
D
,
A
E
,
BF
are its altitudes. The points
E
′
,
F
′
E', F'
E
′
,
F
′
are symetrical of
E
,
F
E, F
E
,
F
with respect accordingly to
A
A
A
and
B
B
B
. The point
C
1
C_1
C
1
lies on
C
D
→
\overrightarrow{CD}
C
D
, such that
D
C
1
=
3
C
D
DC_1=3CD
D
C
1
=
3
C
D
. Prove that
∠
E
′
C
1
F
′
=
∠
A
C
B
\angle E'C_1F'=\angle ACB
∠
E
′
C
1
F
′
=
∠
A
CB
1
1
Hide problems
A cyclic quadrilateral
An convex qudrilateral
A
B
C
D
ABCD
A
BC
D
is given.
O
O
O
is the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. The points
A
1
,
B
1
,
C
1
,
D
1
A_1,B_1,C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
lie respectively on
A
O
,
B
O
,
C
O
,
D
O
AO, BO, CO, DO
A
O
,
BO
,
CO
,
D
O
such that
A
A
1
=
C
C
1
,
B
B
1
=
D
D
1
AA_1=CC_1, BB_1=DD_1
A
A
1
=
C
C
1
,
B
B
1
=
D
D
1
. The circumcircles of
△
A
O
B
\triangle AOB
△
A
OB
and
△
C
O
D
\triangle COD
△
CO
D
meet at second time at
M
M
M
and the the circumcircles of
△
A
O
D
\triangle AOD
△
A
O
D
and
△
B
O
C
\triangle BOC
△
BOC
- at
N
N
N
. The circumcircles of
△
A
1
O
B
1
\triangle A_1OB_1
△
A
1
O
B
1
and
△
C
1
O
D
1
\triangle C_1OD_1
△
C
1
O
D
1
meet at second time at
P
P
P
and the the circumcircles of
△
A
1
O
D
1
\triangle A_1OD_1
△
A
1
O
D
1
and
△
B
1
O
C
1
\triangle B_1OC_1
△
B
1
O
C
1
- at
Q
Q
Q
. Prove that the quadrilateral
M
N
P
Q
MNPQ
MNPQ
is cyclic.
4
1
Hide problems
all triples (p,a,m) satisfying a condition
Find all triples (p,a,m); p is a prime number,
a
,
m
∈
N
a,m\in \mathbb{N}
a
,
m
∈
N
, which satisfy:
a
≤
5
p
2
a\leq 5p^2
a
≤
5
p
2
and
(
p
−
1
)
!
+
a
=
p
m
(p-1)!+a=p^m
(
p
−
1
)!
+
a
=
p
m
.
2
1
Hide problems
numbers written on a board
Let
m
>
1
m>1
m
>
1
be a natural number and
N
=
m
2017
+
1
N=m^{2017}+1
N
=
m
2017
+
1
. On a blackboard, left to right, are written the following numbers:
N
,
N
−
m
,
N
−
2
m
,
…
,
2
m
+
1
,
m
+
1
,
1.
N, N-m, N-2m,\dots, 2m+1,m+1, 1.
N
,
N
−
m
,
N
−
2
m
,
…
,
2
m
+
1
,
m
+
1
,
1.
On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.
5
1
Hide problems
A polynomial with real roots...
Let
n
n
n
be a natural number and
f
(
x
)
f(x)
f
(
x
)
be a polynomial with real coefficients having
n
n
n
different positive real roots. Is it possible the polynomial:
x
(
x
+
1
)
(
x
+
2
)
(
x
+
4
)
f
(
x
)
+
a
x(x+1)(x+2)(x+4)f(x)+a
x
(
x
+
1
)
(
x
+
2
)
(
x
+
4
)
f
(
x
)
+
a
to be presented as the
k
k
k
-th power of a polynomial with real coefficients, for some natural
k
≥
2
k\geq 2
k
≥
2
and real
a
a
a
?
3
1
Hide problems
Coloring the subsets of a set
Let
M
M
M
be a set of
2017
2017
2017
positive integers. For any subset
A
A
A
of
M
M
M
we define
f
(
A
)
:
=
{
x
∈
M
∣
the number of the members of
A
,
x
is multiple of, is odd
}
f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}
f
(
A
)
:=
{
x
∈
M
∣
the number of the members of
A
,
x
is multiple of, is odd
}
. Find the minimal natural number
k
k
k
, satisfying the condition: for any
M
M
M
, we can color all the subsets of
M
M
M
with
k
k
k
colors, such that whenever
A
≠
f
(
A
)
A\neq f(A)
A
=
f
(
A
)
,
A
A
A
and
f
(
A
)
f(A)
f
(
A
)
are colored with different colors.