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National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1970 Bulgaria National Olympiad
1970 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
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f(X)=sum |XA^2| in triangle, maximum and minimum in terms of OG
In space, we are given the points
A
,
B
,
C
A,B,C
A
,
B
,
C
and a sphere with center
O
O
O
and radius
1
1
1
. Find the point
X
X
X
from the sphere for which the sum
f
(
X
)
=
∣
X
A
∣
2
+
∣
X
B
∣
2
+
∣
X
C
∣
2
f(X)=|XA|^2+|XB|^2+|XC|^2
f
(
X
)
=
∣
X
A
∣
2
+
∣
XB
∣
2
+
∣
XC
∣
2
attains its maximal and minimal value. Prove that if the segments
O
A
,
O
B
,
O
C
OA,OB,OC
O
A
,
OB
,
OC
are pairwise perpendicular and
d
d
d
is the distance from the center
O
O
O
to the centroid of the triangle
A
B
C
ABC
A
BC
then:(a) the maximum of
f
(
X
)
f(X)
f
(
X
)
is equal to
9
d
2
+
3
+
6
d
9d^2+3+6d
9
d
2
+
3
+
6
d
; (b) the minimum of
f
(
X
)
f(X)
f
(
X
)
is equal to
9
d
2
+
3
−
6
d
9d^2+3-6d
9
d
2
+
3
−
6
d
.K. Dochev and I. Dimovski
Problem 5
1
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side length of inscribable n-gon, inequality with circumscribed n+1-gon
Prove that for
n
≥
5
n\ge5
n
≥
5
the side of regular inscribable
n
n
n
-gon is bigger than the side of regular
n
+
1
n+1
n
+
1
-gon circumscribed around the same circle and if
n
≤
4
n\le4
n
≤
4
the opposite statement is true.
Problem 4
1
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triangle sequence, centers of squares on sides of previous triangle
Let
δ
0
=
△
A
0
B
0
C
0
\delta_0=\triangle A_0B_0C_0
δ
0
=
△
A
0
B
0
C
0
be a triangle. On each of the sides
B
0
C
0
B_0C_0
B
0
C
0
,
C
0
A
0
C_0A_0
C
0
A
0
,
A
0
B
0
A_0B_0
A
0
B
0
, there are constructed squares in the halfplane, not containing the respective vertex
A
0
,
B
0
,
C
0
A_0,B_0,C_0
A
0
,
B
0
,
C
0
and
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are the centers of the constructed squares. If we use the triangle
δ
1
=
△
A
1
B
1
C
1
\delta_1=\triangle A_1B_1C_1
δ
1
=
△
A
1
B
1
C
1
in the same way we may construct the triangle
δ
2
=
△
A
2
B
2
C
2
\delta_2=\triangle A_2B_2C_2
δ
2
=
△
A
2
B
2
C
2
; from
δ
2
=
△
A
2
B
2
C
2
\delta_2=\triangle A_2B_2C_2
δ
2
=
△
A
2
B
2
C
2
we may construct
δ
3
=
△
A
3
B
3
C
3
\delta_3=\triangle A_3B_3C_3
δ
3
=
△
A
3
B
3
C
3
and etc. Prove that:(a) segments
A
0
A
1
,
B
0
B
1
,
C
0
C
1
A_0A_1,B_0B_1,C_0C_1
A
0
A
1
,
B
0
B
1
,
C
0
C
1
are respectively equal and perpendicular to
B
1
C
1
,
C
1
A
1
,
A
1
B
1
B_1C_1,C_1A_1,A_1B_1
B
1
C
1
,
C
1
A
1
,
A
1
B
1
; (b) vertices
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
of the triangle
δ
1
\delta_1
δ
1
lies respectively over the segments
A
0
A
3
,
B
0
B
3
,
C
0
C
3
A_0A_3,B_0B_3,C_0C_3
A
0
A
3
,
B
0
B
3
,
C
0
C
3
(defined by the vertices of
δ
0
\delta_0
δ
0
and
δ
1
\delta_1
δ
1
) and divide them in ratio
2
:
1
2:1
2
:
1
.K. Dochev
Problem 3
1
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8x8 board with white/black pools, max/min # of mixed pairs
On a chessboard (with
64
64
64
squares) there are situated
32
32
32
white and
32
32
32
black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools.K. Dochev
Problem 2
1
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rates, control car switching between moving bicyclicsts
Two bicyclists traveled the distance from
A
A
A
to
B
B
B
, which is
100
100
100
km, with speed
30
30
30
km/h and it is known that the first started
30
30
30
minutes before the second.
20
20
20
minutes after the start of the first bicyclist from
A
A
A
, there is a control car started whose speed is
90
90
90
km/h and it is known that the car is reached the first bicyclist and is driving together with him for
10
10
10
minutes, went back to the second and was driving for
10
10
10
minutes with him and after that the car is started again to the first bicyclist with speed
90
90
90
km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist?K. Dochev
Problem 1
1
Hide problems
prime divisors of a^6-1 divide either a^2-1 or a^3-1 (Bulgaria 1970 P1)
Find all natural numbers
a
>
1
a>1
a
>
1
, with the property that every prime divisor of
a
6
−
1
a^6-1
a
6
−
1
divides also at least one of the numbers
a
3
−
1
a^3-1
a
3
−
1
,
a
2
−
1
a^2-1
a
2
−
1
.K. Dochev