MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1991 Bulgaria National Olympiad
1991 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
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putting white/black checkers on nxn board
White and black checkers are put on the squares of an
n
×
n
n\times n
n
×
n
chessboard
(
n
≥
2
)
(n\ge2)
(
n
≥
2
)
according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.
Problem 5
1
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triangle based on arc, area minimization
On a unit circle with center
O
O
O
,
A
B
AB
A
B
is an arc with the central angle
α
<
9
0
∘
\alpha<90^\circ
α
<
9
0
∘
. Point
H
H
H
is the foot of the perpendicular from
A
A
A
to
O
B
OB
OB
,
T
T
T
is a point on arc
A
B
AB
A
B
, and
l
l
l
is the tangent to the circle at
T
T
T
. The line
l
l
l
and the angle
A
H
B
AHB
A
H
B
form a triangle
Δ
\Delta
Δ
.(a) Prove that the area of
Δ
\Delta
Δ
is minimal when
T
T
T
is the midpoint of arc
A
B
AB
A
B
. (b) Prove that if
S
α
S_\alpha
S
α
is the minimal area of
Δ
\Delta
Δ
then the function
S
α
α
\frac{S_\alpha}\alpha
α
S
α
has a limit when
α
→
0
\alpha\to0
α
→
0
and find this limit.
Problem 4
1
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P(x)P''(x)≤P'(x)^2 if P has deg P roots
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial of degree
n
n
n
with real coefficients, having
n
n
n
(not necessarily distinct) real roots. Prove that for all real
x
x
x
,
f
(
x
)
f
′
′
(
x
)
≤
f
′
(
x
)
2
.
f(x)f''(x)\le f'(x)^2.
f
(
x
)
f
′′
(
x
)
≤
f
′
(
x
)
2
.
Problem 3
1
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p^3|(kp choose k)-k
Prove that for every prime number
p
≥
5
p\ge5
p
≥
5
,(a)
p
3
p^3
p
3
divides
(
2
p
p
)
−
2
\binom{2p}p-2
(
p
2
p
)
−
2
; (b)
p
3
p^3
p
3
divides
(
k
p
p
)
−
k
\binom{kp}p-k
(
p
k
p
)
−
k
for every natural number
k
k
k
.
Problem 2
1
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cutting cube into n^3 unit cubes
Let
K
K
K
be a cube with edge
n
n
n
, where
n
>
2
n>2
n
>
2
is an even integer. Cube
K
K
K
is divided into
n
3
n^3
n
3
unit cubes. We call any set of
n
2
n^2
n
2
unit cubes lying on the same horizontal or vertical level a layer. We dispose of
n
3
4
\frac{n^3}4
4
n
3
colors, in each of which we paint exactly
4
4
4
unit cubes. Prove that we can always select
n
n
n
unit cubes of distinct colors, no two of which lie on the same layer.
Problem 1
1
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point on triangle altitude, projection onto sides
Let
M
M
M
be a point on the altitude
C
D
CD
C
D
of an acute-angled triangle
A
B
C
ABC
A
BC
, and
K
K
K
and
L
L
L
the orthogonal projections of
M
M
M
on
A
C
AC
A
C
and
B
C
BC
BC
. Suppose that the incenter and circumcenter of the triangle lie on the segment
K
L
KL
K
L
.(a) Prove that
C
D
=
R
+
r
CD=R+r
C
D
=
R
+
r
, where
R
R
R
and
r
r
r
are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio
C
M
:
C
D
CM:CD
CM
:
C
D
.