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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1973 Bulgaria National Olympiad
1973 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 5
1
Hide problems
line passes through incenter iff side equation holds
Let the line
ℓ
\ell
ℓ
intersects the sides
A
C
,
B
C
AC,BC
A
C
,
BC
of the triangle
A
B
C
ABC
A
BC
respectively at the points
E
E
E
and
F
F
F
. Prove that the line
ℓ
\ell
ℓ
is passing through the incenter of the triangle
A
B
C
ABC
A
BC
if and only if the following equality is true:
B
C
⋅
A
E
C
E
+
A
C
⋅
B
F
C
F
=
A
B
.
BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.
BC
⋅
CE
A
E
+
A
C
⋅
CF
BF
=
A
B
.
H. Lesov
Problem 4
1
Hide problems
trig function differentiable at 0, f(x)=g(x)f(x/2) recurrence
Find all functions
f
(
x
)
f(x)
f
(
x
)
defined in the range
(
−
π
2
,
π
2
)
\left(-\frac\pi2,\frac\pi2\right)
(
−
2
π
,
2
π
)
that are differentiable at
0
0
0
and satisfy
f
(
x
)
=
1
2
(
1
+
1
cos
x
)
f
(
x
2
)
f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)
f
(
x
)
=
2
1
(
1
+
cos
x
1
)
f
(
2
x
)
for every
x
x
x
in the range
(
−
π
2
,
π
2
)
\left(-\frac\pi2,\frac\pi2\right)
(
−
2
π
,
2
π
)
.L. Davidov
Problem 3
1
Hide problems
one number has two equal digits in base 10
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
are different integer numbers in the range
[
100
,
200
]
[100,200]
[
100
,
200
]
for which:
a
1
+
a
2
+
…
+
a
n
≥
11100
a_1+a_2+\ldots+a_n\ge11100
a
1
+
a
2
+
…
+
a
n
≥
11100
. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits.L. Davidov
Problem 2
1
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arithmetic sequence as sum of two geometric sequences
Let the numbers
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
form an arithmetic progression with difference
d
≠
0
d\ne0
d
=
0
. Prove that there are no exists geometric progressions
b
1
,
b
2
,
b
3
,
b
4
b_1,b_2,b_3,b_4
b
1
,
b
2
,
b
3
,
b
4
and
c
1
,
c
2
,
c
3
,
c
4
c_1,c_2,c_3,c_4
c
1
,
c
2
,
c
3
,
c
4
such that:
a
1
=
b
1
+
c
1
,
a
2
=
b
2
+
c
2
,
a
3
=
b
3
+
c
3
,
a
4
=
b
4
+
c
4
.
a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.
a
1
=
b
1
+
c
1
,
a
2
=
b
2
+
c
2
,
a
3
=
b
3
+
c
3
,
a
4
=
b
4
+
c
4
.
Problem 6
1
Hide problems
tetrahedrons, minimum sum of volumes
In the tetrahedron
A
B
C
D
ABCD
A
BC
D
,
E
E
E
and
F
F
F
are the midpoints of
B
C
BC
BC
and
A
D
AD
A
D
,
G
G
G
is the midpoint of the segment
E
F
EF
EF
. Construct a plane through
G
G
G
intersecting the segments
A
B
AB
A
B
,
A
C
AC
A
C
,
A
D
AD
A
D
in the points
M
,
N
,
P
M,N,P
M
,
N
,
P
respectively in such a way that the sum of the volumes of the tetrahedrons
B
M
N
P
BMNP
BMNP
,
C
M
N
P
CMNP
CMNP
and
D
M
N
P
DMNP
D
MNP
to be minimal.H. Lesov
Problem 1
1
Hide problems
a_m and a_n relatively prime for m≠n
Let the sequence
a
1
,
a
2
,
…
,
a
n
,
…
a_1,a_2,\ldots,a_n,\ldots
a
1
,
a
2
,
…
,
a
n
,
…
is defined by the conditions:
a
1
=
2
a_1=2
a
1
=
2
and
a
n
+
1
=
a
n
2
−
a
n
+
1
a_{n+1}=a_n^2-a_n+1
a
n
+
1
=
a
n
2
−
a
n
+
1
(
n
=
1
,
2
,
…
)
(n=1,2,\ldots)
(
n
=
1
,
2
,
…
)
. Prove that:(a)
a
m
a_m
a
m
and
a
n
a_n
a
n
are relatively prime numbers when
m
≠
n
m\ne n
m
=
n
. (b)
lim
n
→
∞
∑
k
=
1
n
1
a
k
=
1
\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1
lim
n
→
∞
∑
k
=
1
n
a
k
1
=
1
I. Tonov