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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1965 Bulgaria National Olympiad
1965 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(4)
Problem 4
1
Hide problems
two lines, another _|_ to both in space
In the space there are given crossed lines
s
s
s
and
t
t
t
such that
∠
(
s
,
t
)
=
6
0
∘
\angle(s,t)=60^\circ
∠
(
s
,
t
)
=
6
0
∘
and a segment
A
B
AB
A
B
perpendicular to them. On
A
B
AB
A
B
it is chosen a point
C
C
C
for which
A
C
:
C
B
=
2
:
1
AC:CB=2:1
A
C
:
CB
=
2
:
1
and the points
M
M
M
and
N
N
N
are moving on the lines
s
s
s
and
t
t
t
in such a way that
A
M
=
2
B
N
AM=2BN
A
M
=
2
BN
. The angle between vectors
A
M
→
\overrightarrow{AM}
A
M
and
B
M
→
\overrightarrow{BM}
BM
is
6
0
∘
60^\circ
6
0
∘
. Prove that:(a) the segment
M
N
MN
MN
is perpendicular to
t
t
t
; (b) the plane
α
\alpha
α
, perpendicular to
A
B
AB
A
B
in point
C
C
C
, intersects the plane
C
M
N
CMN
CMN
on fixed line
ℓ
\ell
ℓ
with given direction in respect to
s
s
s
; (c) all planes passing by
e
l
l
ell
e
ll
and perpendicular to
A
B
AB
A
B
intersect the lines
s
s
s
and
t
t
t
respectively at points
M
M
M
and
N
N
N
for which
A
M
=
2
B
N
AM=2BN
A
M
=
2
BN
and
M
N
⊥
t
MN\perp t
MN
⊥
t
.
Problem 3
1
Hide problems
angle bisector intersects circumcircle at X
In the triangle
A
B
C
ABC
A
BC
, angle bisector
C
D
CD
C
D
intersects the circumcircle of
A
B
C
ABC
A
BC
at the point
K
K
K
.(a) Prove the equalities:
1
I
D
−
1
I
K
=
1
C
I
,
C
I
I
D
−
I
D
D
K
=
1
\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1
I
D
1
−
I
K
1
=
C
I
1
,
I
D
C
I
−
DK
I
D
=
1
where
I
I
I
is the center of the inscribed circle of triangle
A
B
C
ABC
A
BC
. (b) On the segment
C
K
CK
C
K
some point
P
P
P
is chosen whose projections on
A
C
,
B
C
,
A
B
AC,BC,AB
A
C
,
BC
,
A
B
respectively are
P
1
,
P
2
,
P
3
P_1,P_2,P_3
P
1
,
P
2
,
P
3
. The lines
P
P
3
PP_3
P
P
3
and
P
1
P
2
P_1P_2
P
1
P
2
intersect at a point
M
M
M
. Find the locus of
M
M
M
when
P
P
P
moves around segment
C
K
CK
C
K
.
Problem 2
1
Hide problems
trigonometric inequality with natural parameter
Prove the inequality:
(
1
+
sin
2
α
)
n
+
(
1
+
cos
2
α
)
n
≥
2
(
3
2
)
n
(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n
(
1
+
sin
2
α
)
n
+
(
1
+
cos
2
α
)
n
≥
2
(
2
3
)
n
is true for every natural number
n
n
n
. When does equality hold?
Problem 1
1
Hide problems
a nice type of integer triples (Bulgaria 1965 P1)
The numbers
2
,
3
,
7
2,3,7
2
,
3
,
7
have the property that the product of any two of them increased by
1
1
1
is divisible by the third number. Prove that this triple of integer numbers greater than
1
1
1
is the only triple with the given property.