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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Autumn Mathematical Competition
2007 Bulgarian Autumn Math Competition
2007 Bulgarian Autumn Math Competition
Part of
Bulgarian Autumn Mathematical Competition
Subcontests
(20)
Problem 12.4
1
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linear recurrence sequence with prime coefficients
Let
p
p
p
and
q
q
q
be prime numbers and
{
a
n
}
n
=
1
∞
\{a_{n}\}_{n=1}^{\infty}
{
a
n
}
n
=
1
∞
be a sequence of integers defined by: a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n} \forall n\geq 0 Find
p
p
p
and
q
q
q
if there exists an integer
k
k
k
such that
a
3
k
=
−
3
a_{3k}=-3
a
3
k
=
−
3
.
Problem 12.3
1
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Symmetric 3-var inequality
Find all real numbers
r
r
r
, such that the inequality
r
(
a
b
+
b
c
+
c
a
)
+
(
3
−
r
)
(
1
a
+
1
b
+
1
c
)
≥
9
r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9
r
(
ab
+
b
c
+
c
a
)
+
(
3
−
r
)
(
a
1
+
b
1
+
c
1
)
≥
9
holds for any real
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
.
Problem 12.2
1
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Intersection of plane and a pyramid is a trapezoid
All edges of the triangular pyramid
A
B
C
D
ABCD
A
BC
D
are equal in length. Let
M
M
M
be the midpoint of
D
B
DB
D
B
,
N
N
N
is the point on
A
B
‾
\overline{AB}
A
B
, such that
2
N
A
=
N
B
2NA=NB
2
N
A
=
NB
and
N
∉
A
B
N\not\in AB
N
∈
A
B
and
P
P
P
is a point on the altitude through point
D
D
D
in
△
B
C
D
\triangle BCD
△
BC
D
. Find
∠
M
P
D
\angle MPD
∠
MP
D
if the intersection of the pyramid with the plane
(
N
M
P
)
(NMP)
(
NMP
)
is a trapezoid.
Problem 12.1
1
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Parametric trigonometric equation
Determine the values of the real parameter
a
a
a
, such that the equation
sin
2
x
sin
4
x
−
sin
x
sin
3
x
=
a
\sin 2x\sin 4x-\sin x\sin 3x=a
sin
2
x
sin
4
x
−
sin
x
sin
3
x
=
a
has a unique solution in the interval
[
0
,
π
)
[0,\pi)
[
0
,
π
)
.
Problem 11.4
1
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Graph stays connected after deleting a vertex
There are 1000 towns
A
1
,
A
2
,
…
,
A
1000
A_{1},A_{2},\ldots ,A_{1000}
A
1
,
A
2
,
…
,
A
1000
with airports in a country and some of them are connected via flights. It's known that the
i
i
i
-th town is connected with
d
i
d_{i}
d
i
other towns where
d
1
≤
d
2
≤
…
≤
d
1000
d_{1}\leq d_{2}\leq \ldots \leq d_{1000}
d
1
≤
d
2
≤
…
≤
d
1000
and
d
j
≥
j
+
1
d_{j}\geq j+1
d
j
≥
j
+
1
for every
j
=
1
,
2
,
…
999
−
d
999
j=1,2,\ldots 999-d_{999}
j
=
1
,
2
,
…
999
−
d
999
. Prove that if the airport of any town
A
k
A_{k}
A
k
is closed, then we'd still be able to get from any town
A
i
A_{i}
A
i
to any
A
j
A_{j}
A
j
for
i
,
j
≠
k
i,j\neq k
i
,
j
=
k
(possibly by more than one flight).
Problem 11.3
1
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Equal segments imply angle bisector
In
△
A
B
C
\triangle ABC
△
A
BC
we have that
C
C
1
CC_{1}
C
C
1
is an angle bisector. The points
P
∈
C
1
B
P\in C_{1}B
P
∈
C
1
B
,
Q
∈
B
C
Q\in BC
Q
∈
BC
,
R
∈
A
C
R\in AC
R
∈
A
C
,
S
∈
A
C
1
S\in AC_{1}
S
∈
A
C
1
satisfy
C
1
P
=
P
Q
=
Q
C
C_{1}P=PQ=QC
C
1
P
=
PQ
=
QC
and
C
R
=
R
S
=
S
C
1
CR=RS=SC_{1}
CR
=
RS
=
S
C
1
. Prove that
C
C
1
CC_{1}
C
C
1
bisects
∠
S
C
P
\angle SCP
∠
SCP
.
Problem 11.2
1
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Inequality with a parameter
Find all values of the parameter
a
a
a
for which the inequality
x
−
x
2
−
a
+
6
a
−
2
x
−
x
2
≤
10
a
−
2
x
−
4
x
2
\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}
x
−
x
2
−
a
+
6
a
−
2
x
−
x
2
≤
10
a
−
2
x
−
4
x
2
has a unique solution.
Problem 11.1
1
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Trigonometric equation
Let
0
<
α
,
β
<
π
2
0<\alpha,\beta<\frac{\pi}{2}
0
<
α
,
β
<
2
π
which satisfy
(
cos
2
α
+
cos
2
β
)
(
1
+
tan
α
tan
β
)
=
2
(\cos^2\alpha+\cos^2\beta)(1+\tan\alpha\tan\beta)=2
(
cos
2
α
+
cos
2
β
)
(
1
+
tan
α
tan
β
)
=
2
Prove that
α
+
β
=
π
2
\alpha+\beta=\frac{\pi}{2}
α
+
β
=
2
π
.
Problem 10.4
1
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Table of zeros and ones with special property
Find all pairs of natural numbers
(
m
,
n
)
(m,n)
(
m
,
n
)
,
m
≤
n
m\leq n
m
≤
n
, such that there exists a table with
m
m
m
rows and
n
n
n
columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.
Problem 10.3
1
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Sum of numbers less than x and coprime to x
For a natural number
m
>
1
m>1
m
>
1
we'll denote with
f
(
m
)
f(m)
f
(
m
)
the sum of all natural numbers less than
m
m
m
, which are also coprime to
m
m
m
. Find all natural numbers
n
n
n
, such that there exist natural numbers
k
k
k
and
ℓ
\ell
ℓ
which satisfy
f
(
n
k
)
=
n
ℓ
f(n^{k})=n^{\ell}
f
(
n
k
)
=
n
ℓ
.
Problem 10.2
1
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Classic geo
Let
A
C
>
B
C
AC>BC
A
C
>
BC
in
△
A
B
C
\triangle ABC
△
A
BC
and
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
respectively. The angle bisector of
∠
B
\angle B
∠
B
intersects
M
N
‾
\overline{MN}
MN
at
P
P
P
. The incircle of
△
A
B
C
\triangle ABC
△
A
BC
has center
I
I
I
and touches
B
C
BC
BC
at
Q
Q
Q
. The perpendiculars from
P
P
P
and
Q
Q
Q
to
M
N
MN
MN
and
B
C
BC
BC
respectively intersect at
R
R
R
. Let
S
=
A
B
∩
R
N
S=AB\cap RN
S
=
A
B
∩
RN
. a) Prove that
P
C
Q
I
PCQI
PCQ
I
is cyclic b) Express the length of the segment
B
S
BS
BS
with
a
a
a
,
b
b
b
,
c
c
c
- the side lengths of
△
A
B
C
\triangle ABC
△
A
BC
.
Problem 10.1
1
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Roots of quadratic satisfy equation
Find all integers
b
b
b
and
c
c
c
for which the equation
x
2
−
b
x
+
c
=
0
x^2-bx+c=0
x
2
−
b
x
+
c
=
0
has two real roots
x
1
x_{1}
x
1
and
x
2
x_{2}
x
2
satisfying
x
1
2
+
x
2
2
=
5
x_{1}^2+x_{2}^2=5
x
1
2
+
x
2
2
=
5
.
Problem 9.4
1
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Divisor of specific form
Find the smallest natural number, which divides
2
n
+
15
2^{n}+15
2
n
+
15
for some natural number
n
n
n
and can be expressed in the form
3
x
2
−
4
x
y
+
3
y
2
3x^2-4xy+3y^2
3
x
2
−
4
x
y
+
3
y
2
for some integers
x
x
x
and
y
y
y
.
Problem 9.3
1
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Cyclic quads wanted
Let the intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
be point
E
E
E
. Let
M
M
M
be the midpoint of
A
E
AE
A
E
and
N
N
N
be the midpoint of
C
D
CD
C
D
. It's known that
B
D
BD
B
D
bisects
∠
A
B
C
\angle ABC
∠
A
BC
. Prove that
A
B
C
D
ABCD
A
BC
D
is cyclic if and only if
M
B
C
N
MBCN
MBCN
is cyclic.
Problem 9.2
1
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Algebra manipulation excercise
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers, such that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
and
a
4
+
b
4
+
c
4
=
50
a^4+b^4+c^4=50
a
4
+
b
4
+
c
4
=
50
. Determine the value of
a
b
+
b
c
+
c
a
ab+bc+ca
ab
+
b
c
+
c
a
.
Problem 9.1
1
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Graphs of functions
We're given the functions
f
(
x
)
=
∣
x
−
1
∣
−
∣
x
−
2
∣
f(x)=|x-1|-|x-2|
f
(
x
)
=
∣
x
−
1∣
−
∣
x
−
2∣
and
g
(
x
)
=
∣
x
−
3
∣
g(x)=|x-3|
g
(
x
)
=
∣
x
−
3∣
. a) Draw the graph of the function
f
(
x
)
f(x)
f
(
x
)
. b) Determine the area of the section enclosed by the functions
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
.
Problem 8.4
1
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Intersecting lines in a heptagon
Let
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
be a regular heptagon. We'll call the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
E
DE
D
E
,
E
F
EF
EF
,
F
G
FG
FG
and
G
A
GA
G
A
opposite to the vertices
E
E
E
,
F
F
F
,
G
G
G
,
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
respectively. If
M
M
M
is a point inside the heptagon, we'll say that the line through
M
M
M
and a vertex of the heptagon intersects a side of it (without the vertices) at a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
e
r
f
e
c
t
<
/
s
p
a
n
>
<span class='latex-italic'>perfect</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
er
f
ec
t
<
/
s
p
an
>
point, if this side is opposite the vertex. Prove that for every choice of
M
M
M
, the number of
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
e
r
f
e
c
t
<
/
s
p
a
n
>
<span class='latex-italic'>perfect</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
er
f
ec
t
<
/
s
p
an
>
points is always odd.
Problem 8.3
1
Hide problems
Expression of primes is a square
Determine all triplets of prime numbers
p
<
q
<
r
p<q<r
p
<
q
<
r
, such that
p
+
q
=
r
p+q=r
p
+
q
=
r
and
(
r
−
p
)
(
q
−
p
)
−
27
p
(r-p)(q-p)-27p
(
r
−
p
)
(
q
−
p
)
−
27
p
is a square.
Problem 8.2
1
Hide problems
Area related locus of a point
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Determine all points
M
M
M
, which lie inside
A
B
C
D
ABCD
A
BC
D
, such that the areas of
A
B
C
M
ABCM
A
BCM
and
A
M
C
D
AMCD
A
MC
D
are equal.
Problem 8.1
1
Hide problems
System of equations with a parameter
Determine all real
a
a
a
, such that the solutions to the system of equations
{
3
x
−
5
3
+
3
x
+
5
4
≥
x
7
−
1
15
(
2
x
−
a
)
3
+
(
2
x
+
a
)
(
1
−
4
x
2
)
+
16
x
2
a
−
6
x
2
a
+
a
3
≤
2
a
2
+
a
\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}
{
3
3
x
−
5
+
4
3
x
+
5
≥
7
x
−
15
1
(
2
x
−
a
)
3
+
(
2
x
+
a
)
(
1
−
4
x
2
)
+
16
x
2
a
−
6
x
2
a
+
a
3
≤
2
a
2
+
a
form an interval with length
32
225
\frac{32}{225}
225
32
.