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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1964 Bulgaria National Olympiad
1964 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(4)
Problem 4
1
Hide problems
three lines with each pair crossing, triangles congruent
Let
a
1
,
b
1
,
c
1
a_1,b_1,c_1
a
1
,
b
1
,
c
1
are three lines each two of them are mutually crossed and aren't parallel to some plane. The lines
a
2
,
b
2
,
c
2
a_2,b_2,c_2
a
2
,
b
2
,
c
2
intersect the lines
a
1
,
b
1
,
c
1
a_1,b_1,c_1
a
1
,
b
1
,
c
1
at the points
a
2
a_2
a
2
in
A
A
A
,
C
2
C_2
C
2
,
B
1
B_1
B
1
;
b
2
b_2
b
2
in
C
1
C_1
C
1
,
B
B
B
,
A
2
A_2
A
2
;
c
2
c_2
c
2
in
B
2
B_2
B
2
,
A
1
A_1
A
1
,
C
C
C
respectively in such a way that
A
A
A
is the perpendicular bisector of
B
1
C
2
B_1C_2
B
1
C
2
,
B
B
B
is the perpendicular bisector of
C
1
A
2
C_1A_2
C
1
A
2
and
C
C
C
is the perpendicular bisector of
A
1
B
2
A_1B_2
A
1
B
2
. Prove that:(a)
A
A
A
is the perpendicular bisector of
B
2
C
1
B_2C_1
B
2
C
1
,
B
B
B
is the perpendicular bisector of
C
2
A
1
C_2A_1
C
2
A
1
and
C
C
C
is the perpendicular bisector of
A
2
B
1
A_2B_1
A
2
B
1
; (b) triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
are the same.
Problem 3
1
Hide problems
locus for which three points collinear
There are given two intersecting lines
g
1
,
g
2
g_1,g_2
g
1
,
g
2
and a point
P
P
P
in their plane such that
∠
(
g
1
,
g
2
)
≠
9
0
∘
\angle(g1,g2)\ne90^\circ
∠
(
g
1
,
g
2
)
=
9
0
∘
. Its symmetrical points on any point
M
M
M
in the same plane with respect to the given lines are
M
1
M_1
M
1
and
M
2
M_2
M
2
. Prove that:(a) the locus of the point
M
M
M
for which the points
M
1
,
M
2
M_1,M_2
M
1
,
M
2
and
P
P
P
lie on a common line is a circle
k
k
k
passing through the intersection point of
g
1
g_1
g
1
and
g
2
g_2
g
2
. (b) the point
P
P
P
is an orthocenter of a triangle, inscribed in the circle
k
k
k
whose sides lie at the lines
g
1
g_1
g
1
and
g
2
g_2
g
2
.
Problem 2
1
Hide problems
system in n variables, R (Bulgaria 1964 P2)
Find all
n
n
n
-tuples of reals
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
satisfying the system:
{
x
1
x
2
⋯
x
n
=
1
x
1
−
x
2
x
3
⋯
x
n
=
1
x
1
x
2
−
x
3
x
4
⋯
x
n
=
1
⋮
x
1
x
2
⋯
x
n
−
1
−
x
n
=
1
\begin{cases}x_1x_2\cdots x_n=1\\x_1-x_2x_3\cdots x_n=1\\x_1x_2-x_3x_4\cdots x_n=1\\\vdots\\x_1x_2\cdots x_{n-1}-x_n=1\end{cases}
⎩
⎨
⎧
x
1
x
2
⋯
x
n
=
1
x
1
−
x
2
x
3
⋯
x
n
=
1
x
1
x
2
−
x
3
x
4
⋯
x
n
=
1
⋮
x
1
x
2
⋯
x
n
−
1
−
x
n
=
1
Problem 1
1
Hide problems
digit shift preserves 7-divisibility (Bulgaria 1964 P1)
A
6
n
6n
6
n
-digit number is divisible by
7
7
7
. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by
7
7
7
.