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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1967 Bulgaria National Olympiad
1967 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(4)
Problem 4
1
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point not on triangle plane, tetrahedron ABCD with heights drawn
Outside of the plane of the triangle
A
B
C
ABC
A
BC
is given point
D
D
D
. (a) prove that if the segment
D
A
DA
D
A
is perpendicular to the plane
A
B
C
ABC
A
BC
then orthogonal projection of the orthocenter of the triangle
A
B
C
ABC
A
BC
on the plane
B
C
D
BCD
BC
D
coincides with the orthocenter of the triangle
B
C
D
BCD
BC
D
. (b) for all tetrahedrons
A
B
C
D
ABCD
A
BC
D
with base, the triangle
A
B
C
ABC
A
BC
with smallest of the four heights that from the vertex
D
D
D
, find the locus of the foot of that height.
Problem 3
1
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right angle triangle with circumcircle, Toricelli point lies on circumcircle
It is given a right-angled triangle
A
B
C
ABC
A
BC
and its circumcircle
k
k
k
. (a) prove that the radii of the circle
k
1
k_1
k
1
tangent to the cathets of the triangle and to the circle
k
k
k
is equal to the diameter of the incircle of the triangle ABC. (b) on the circle
k
k
k
there may be found a point
M
M
M
for which the sum
M
A
+
M
B
+
M
C
MA+MB+MC
M
A
+
MB
+
MC
is as large as possible.
Problem 2
1
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bounding (y+1)^n-y^n, and a weaker FLT condition (Bulgaria 1967 P2)
Prove that: (a) if
y
<
1
2
y<\frac12
y
<
2
1
and
n
≥
3
n\ge3
n
≥
3
is a natural number then
(
y
+
1
)
n
≥
y
n
+
(
1
+
2
y
)
n
2
(y+1)^n\ge y^n+(1+2y)^\frac n2
(
y
+
1
)
n
≥
y
n
+
(
1
+
2
y
)
2
n
; (b) if
x
,
y
,
z
x,y,z
x
,
y
,
z
and
n
≥
3
n\ge3
n
≥
3
are natural numbers for which
x
2
−
1
≤
2
y
x^2-1\le2y
x
2
−
1
≤
2
y
then
x
n
+
y
n
≠
z
n
x^n+y^n\ne z^n
x
n
+
y
n
=
z
n
.
Problem 1
1
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given solution to xy-xz+yt=182 (Bulgaria 1967 P1)
The numbers
12
,
14
,
37
,
65
12,14,37,65
12
,
14
,
37
,
65
are one of the solutions of the equation
x
y
−
x
z
+
y
t
=
182
xy-xz+yt=182
x
y
−
x
z
+
y
t
=
182
. What number corresponds to which letter?