MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1978 Bulgaria National Olympiad
1978 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 5
1
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a nice property of polygons, circumcircle existence for some 3 vertices
Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.Jordan Tabov
Problem 4
1
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min/max of ((a+b+c)^2)/bc
Find the greatest possible real value of
S
S
S
and smallest possible value of
T
T
T
such that for every triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
(
a
≤
b
≤
c
)
(a\le b\le c)
(
a
≤
b
≤
c
)
to be true the inequalities:
S
≤
(
a
+
b
+
c
)
2
b
c
≤
T
.
S\le\frac{(a+b+c)^2}{bc}\le T.
S
≤
b
c
(
a
+
b
+
c
)
2
≤
T
.
Problem 6
1
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5-gon pyramid, longest edge inequality
The base of the pyramid with vertex
S
S
S
is a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
for which
B
C
>
D
E
BC>DE
BC
>
D
E
and
A
B
>
C
D
AB>CD
A
B
>
C
D
. If
A
S
AS
A
S
is the longest edge of the pyramid prove that
B
S
>
C
S
BS>CS
BS
>
CS
.Jordan Tabov
Problem 3
1
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5 people sitting around table
On the name day of a man there are
5
5
5
people. The men observed that of any
3
3
3
people there are
2
2
2
that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows.N. Nenov, N. Hazhiivanov
Problem 2
1
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locus, point moves along arc
k
1
k_1
k
1
denotes one of the arcs formed by intersection of the circumference
k
k
k
and the chord
A
B
AB
A
B
.
C
C
C
is the middle point of
k
1
k_1
k
1
. On the half line (ray)
P
C
PC
PC
is drawn the segment
P
M
PM
PM
. Find the locus formed from the point
M
M
M
when
P
P
P
is moving on
k
1
k_1
k
1
.G. Ganchev
Problem 1
1
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sequence is in N
We are given the sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
, for which:
a
n
=
a
n
−
1
2
+
c
a
n
−
2
for all
n
>
2.
a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.
a
n
=
a
n
−
2
a
n
−
1
2
+
c
for all
n
>
2.
Prove that the numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
and
a
1
2
+
a
2
2
+
c
a
1
a
2
\frac{a_1^2+a_2^2+c}{a_1a_2}
a
1
a
2
a
1
2
+
a
2
2
+
c
are whole numbers.