MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1990 Bulgaria National Olympiad
1990 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
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tetrahedron, RS≥2V, S=area of base
The base
A
B
C
ABC
A
BC
of a tetrahedron
M
A
B
C
MABC
M
A
BC
is an equilateral triangle, and the lateral edges
M
A
,
M
B
,
M
C
MA,MB,MC
M
A
,
MB
,
MC
are sides of a triangle of the area
S
S
S
. If
R
R
R
is the circumradius and
V
V
V
the volume of the tetrahedron, prove that
R
S
≥
2
V
RS\ge2V
RS
≥
2
V
. When does equality hold?
Problem 5
1
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triangle in circular arc
Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.
Problem 4
1
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NT set, x+y in M implies x or y in M
Suppose
M
M
M
is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in
M
M
M
, one of these two numbers is in
M
M
M
as well. Prove that the elements of any finite set of natural numbers not belonging to
M
M
M
have a common divisor greater than
1
1
1
.
Problem 3
1
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expression is Z[x], function
Let
n
=
p
1
p
2
⋯
p
s
n=p_1p_2\cdots p_s
n
=
p
1
p
2
⋯
p
s
, where
p
1
,
…
,
p
s
p_1,\ldots,p_s
p
1
,
…
,
p
s
are distinct odd prime numbers. (a) Prove that the expression
F
n
(
x
)
=
∏
(
x
n
p
i
1
⋯
p
i
k
−
1
)
(
−
1
)
k
,
F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},
F
n
(
x
)
=
∏
(
x
p
i
1
⋯
p
i
k
n
−
1
)
(
−
1
)
k
,
where the product goes over all subsets
{
p
i
1
,
…
,
p
i
k
}
\{p_{i_1},\ldots,p_{i_k}\}
{
p
i
1
,
…
,
p
i
k
}
or
{
p
1
,
…
,
p
s
}
\{p_1,\ldots,p_s\}
{
p
1
,
…
,
p
s
}
(including itself and the empty set), can be written as a polynomial in
x
x
x
with integer coefficients. (b) Prove that if
p
p
p
is a prime divisor of
F
n
(
2
)
F_n(2)
F
n
(
2
)
, then either
p
∣
n
p\mid n
p
∣
n
or
n
∣
p
−
1
n\mid p-1
n
∣
p
−
1
.
Problem 2
1
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parabola y=ax^2, OA _|_ OB
Let be given a real number
α
≠
0
\alpha\ne0
α
=
0
. Show that there is a unique point
P
P
P
in the coordinate plane, such that for every line through
P
P
P
which intersects the parabola
y
=
α
x
2
y=\alpha x^2
y
=
α
x
2
in two distinct points
A
A
A
and
B
B
B
, segments
O
A
OA
O
A
and
O
B
OB
OB
are perpendicular (where
O
O
O
is the origin).
Problem 1
1
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removing parity indices from [1990]
Consider the number obtained by writing the numbers
1
,
2
,
…
,
1990
1,2,\ldots,1990
1
,
2
,
…
,
1990
one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?