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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1962 Bulgaria National Olympiad
1962 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(4)
Problem 4
1
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point in triangle, distance to sides and vertices (Bulgaria 1962 P4)
There are given a triangle and some internal point
P
P
P
.
x
,
y
,
z
x,y,z
x
,
y
,
z
are distances from
P
P
P
to the vertices
A
,
B
A,B
A
,
B
and
C
C
C
.
p
,
q
,
r
p,q,r
p
,
q
,
r
are distances from
P
P
P
to the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Prove that:
x
y
z
≥
(
q
+
r
)
(
r
+
p
)
(
p
+
q
)
.
xyz\ge(q+r)(r+p)(p+q).
x
yz
≥
(
q
+
r
)
(
r
+
p
)
(
p
+
q
)
.
Problem 1
1
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y=(x^2-2x+1)/(x^2-2x+2), image of y
It is given the expression
y
=
x
2
−
2
x
+
1
x
2
−
2
x
+
2
y=\frac{x^2-2x+1}{x^2-2x+2}
y
=
x
2
−
2
x
+
2
x
2
−
2
x
+
1
, where
x
x
x
is a variable. Prove that:(a) if
x
1
x_1
x
1
and
x
2
x_2
x
2
are two values of
x
x
x
, the
y
1
y_1
y
1
and
y
2
y_2
y
2
are the respective values of
y
y
y
only if
x
1
<
x
2
x_1<x_2
x
1
<
x
2
,
y
1
<
y
2
y_1<y_2
y
1
<
y
2
; (b) when
x
x
x
is varying
y
y
y
attains all possible values for which
0
≤
y
<
1
0\le y<1
0
≤
y
<
1
.
Problem 3
1
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cross-section of cube _|_ to diagonals (Bulgaria 1962 P3)
It is given a cube with sidelength
a
a
a
. Find the surface of the intersection of the cube with a plane, perpendicular to one of its diagonals and whose distance from the centre of the cube is equal to
h
h
h
.
Problem 2
1
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two diameters of circle, points concyclic (Bulgaria 1962 P2)
It is given a circle with center
O
O
O
and radius
r
r
r
.
A
B
AB
A
B
and
M
N
MN
MN
are two diameters. The lines
M
B
MB
MB
and
N
B
NB
NB
are tangent to the circle at the points
M
′
M'
M
′
and
N
′
N'
N
′
and intersect at point
A
A
A
.
M
′
′
M''
M
′′
and
N
′
′
N''
N
′′
are the midpoints of the segments
A
M
′
AM'
A
M
′
and
A
N
′
AN'
A
N
′
. Prove that:(a) the points
M
,
N
,
N
′
,
M
′
M,N,N',M'
M
,
N
,
N
′
,
M
′
are concyclic. (b) the heights of the triangle
M
′
′
N
′
′
B
M''N''B
M
′′
N
′′
B
intersect in the midpoint of the radius
O
A
OA
O
A
.