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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2010 Bulgaria National Olympiad
2010 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
2
2
Hide problems
The area and the graph - [Bulgaria NMO 2010]
Each of two different lines parallel to the the axis
O
x
Ox
O
x
have exactly two common points on the graph of the function
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f(x)=x^3+ax^2+bx+c
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
. Let
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
be two lines parallel to
O
x
Ox
O
x
axis which meet the graph of
f
f
f
in points
K
1
,
K
2
K_1, K_2
K
1
,
K
2
and
K
3
,
K
4
K_3, K_4
K
3
,
K
4
, respectively. Prove that the quadrilateral formed by
K
1
,
K
2
,
K
3
K_1, K_2, K_3
K
1
,
K
2
,
K
3
and
K
4
K_4
K
4
is a rhombus if and only if its area is equal to
6
6
6
units.
Prove that f(n+f(n))=n for each n - [Bulgaria NMO 2010]
Let
f
:
N
→
N
f: \mathbb N \to \mathbb N
f
:
N
→
N
be a function such that
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and f(n)=n - f(f(n-1)), \forall n \geq 2. Prove that
f
(
n
+
f
(
n
)
)
=
n
f(n+f(n))=n
f
(
n
+
f
(
n
))
=
n
for each positive integer
n
.
n.
n
.
1
2
Hide problems
Winning Strategy For The Game - [Bulgaria NMO 2010]
A table
2
×
2010
2 \times 2010
2
×
2010
is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal
2
×
1
2 \times 1
2
×
1
domino that covers exactly two unit table cells. Then Peter puts vertical
1
×
2
1 \times 2
1
×
2
domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
Does there exist such n? - [Bulgaria NMO 2010]
Does there exist a number
n
=
a
1
a
2
a
3
a
4
a
5
a
6
‾
n=\overline{a_1a_2a_3a_4a_5a_6}
n
=
a
1
a
2
a
3
a
4
a
5
a
6
such that
a
1
a
2
a
3
‾
+
4
=
a
4
a
5
a
6
‾
\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}
a
1
a
2
a
3
+
4
=
a
4
a
5
a
6
(all bases are
10
10
10
) and
n
=
a
k
n=a^k
n
=
a
k
for some positive integers
a
,
k
a,k
a
,
k
with
k
≥
3
?
k \geq 3 \ ?
k
≥
3
?
3
2
Hide problems
Find all possible values of n - [Bulgaria NMO 2010]
Let
a
0
,
a
1
,
…
,
a
9
a_0, a_1, \ldots, a_9
a
0
,
a
1
,
…
,
a
9
and
b
1
,
b
2
,
…
,
b
9
b_1 , b_2, \ldots,b_9
b
1
,
b
2
,
…
,
b
9
be positive integers such that
a
9
<
b
9
a_9<b_9
a
9
<
b
9
and
a
k
≠
b
k
,
1
≤
k
≤
8.
a_k \neq b_k, 1 \leq k \leq 8.
a
k
=
b
k
,
1
≤
k
≤
8.
In a cash dispenser/automated teller machine/ATM there are
n
≥
a
9
n\geq a_9
n
≥
a
9
levs (Bulgarian national currency) and for each
1
≤
i
≤
9
1 \leq i \leq 9
1
≤
i
≤
9
we can take
a
i
a_i
a
i
levs from the ATM (if in the bank there are at least
a
i
a_i
a
i
levs). Immediately after that action the bank puts
b
i
b_i
b
i
levs in the ATM or we take
a
0
a_0
a
0
levs. If we take
a
0
a_0
a
0
levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of
n
n
n
such that after finite number of takings money from the ATM there will be no money in it.
Prove that M ≡ D - [Bulgaria NMO 2010]
Let
k
k
k
be the circumference of the triangle
A
B
C
.
ABC.
A
BC
.
The point
D
D
D
is an arbitrary point on the segment
A
B
.
AB.
A
B
.
Let
I
I
I
and
J
J
J
be the centers of the circles which are tangent to the side
A
B
,
AB,
A
B
,
the segment
C
D
CD
C
D
and the circle
k
.
k.
k
.
We know that the points
A
,
B
,
I
A, B, I
A
,
B
,
I
and
J
J
J
are concyclic. The excircle of the triangle
A
B
C
ABC
A
BC
is tangent to the side
A
B
AB
A
B
in the point
M
.
M.
M
.
Prove that
M
≡
D
.
M \equiv D.
M
≡
D
.