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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1972 Bulgaria National Olympiad
1972 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 4
1
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two points in circle, distance ≥r√2
Find maximal possible number of points lying on or inside a circle with radius
R
R
R
in such a way that the distance between every two points is greater than
R
2
R\sqrt2
R
2
.H. Lesov
Problem 5
1
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quadrilateral with _|_ diagonals, perpendiculars from intersection point
In a circle with radius
R
R
R
, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.(a) Prove that the feet of these perpendiculars
P
1
,
P
2
,
P
3
,
P
4
P_1,P_2,P_3,P_4
P
1
,
P
2
,
P
3
,
P
4
are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities
2
r
1
≤
2
R
1
≤
R
2r_1\le\sqrt2 R_1\le R
2
r
1
≤
2
R
1
≤
R
where
R
1
R_1
R
1
and
r
1
r_1
r
1
are radii respectively of the circumcircle and inscircle to the quadrilateral
P
1
P
2
P
3
P
4
P_1P_2P_3P_4
P
1
P
2
P
3
P
4
. When does equality hold?H. Lesov
Problem 3
1
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trigonometric equality
Prove the equality:
∑
k
=
1
n
−
1
1
sin
2
(
2
k
+
1
)
π
2
n
=
n
2
\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2
k
=
1
∑
n
−
1
sin
2
2
n
(
2
k
+
1
)
π
1
=
n
2
where
n
n
n
is a natural number.H. Lesov
Problem 1
1
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9|(x+a)(x+b)(x+c)-x^3-1 doesn't hold for some x and all a,b,c
Prove that there are don't exist integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that for every integer
x
x
x
the number
A
=
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
−
x
3
−
1
A=(x+a)(x+b)(x+c)-x^3-1
A
=
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
−
x
3
−
1
is divisible by
9
9
9
.I. Tonov
Problem 6
1
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orthocentric tetrahedron, strict inequality
It is given a tetrahedron
A
B
C
D
ABCD
A
BC
D
for which two points of opposite edges are mutually perpendicular. Prove that:(a) the four altitudes of
A
B
C
D
ABCD
A
BC
D
intersects at a common point
H
H
H
; (b)
A
H
+
B
H
+
C
H
+
D
H
<
p
+
2
R
AH+BH+CH+DH<p+2R
A
H
+
B
H
+
C
H
+
DH
<
p
+
2
R
, where
p
p
p
is the sum of the lengths of all edges of
A
B
C
D
ABCD
A
BC
D
and
R
R
R
is the radii of the sphere circumscribed around
A
B
C
D
ABCD
A
BC
D
.H. Lesov
Problem 2
1
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system in R+, four equations/variables
Solve the system of equations:
{
y
(
t
−
y
)
t
−
x
−
4
x
+
z
(
t
−
z
)
t
−
x
−
4
x
=
x
z
(
t
−
z
)
t
−
y
−
4
y
+
x
(
t
−
x
)
t
−
y
−
4
y
=
y
x
(
t
−
x
)
t
−
z
−
4
z
+
y
(
t
−
y
)
t
−
z
−
4
z
=
z
x
+
y
+
z
=
2
t
\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}
⎩
⎨
⎧
t
−
x
y
(
t
−
y
)
−
x
4
+
t
−
x
z
(
t
−
z
)
−
x
4
=
x
t
−
y
z
(
t
−
z
)
−
y
4
+
t
−
y
x
(
t
−
x
)
−
y
4
=
y
t
−
z
x
(
t
−
x
)
−
z
4
+
t
−
z
y
(
t
−
y
)
−
z
4
=
z
x
+
y
+
z
=
2
t
if the following conditions are satisfied:
0
<
x
<
t
0<x<t
0
<
x
<
t
,
0
<
y
<
t
0<y<t
0
<
y
<
t
,
0
<
z
<
t
0<z<t
0
<
z
<
t
.H. Lesov