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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2001 Bulgaria National Olympiad
2001 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
2
2
Hide problems
System of equations!
Find all real values
t
t
t
for which there exist real numbers
x
x
x
,
y
y
y
,
z
z
z
satisfying :
3
x
2
+
3
x
z
+
z
2
=
1
3x^2 + 3xz + z^2 = 1
3
x
2
+
3
x
z
+
z
2
=
1
,
3
y
2
+
3
y
z
+
z
2
=
4
3y^2 + 3yz + z^2 = 4
3
y
2
+
3
yz
+
z
2
=
4
,
x
2
−
x
y
+
y
2
=
t
x^2 - xy + y^2 = t
x
2
−
x
y
+
y
2
=
t
.
A good Parallelogram
Suppose that
A
B
C
D
ABCD
A
BC
D
is a parallelogram such that
D
A
B
>
90
DAB>90
D
A
B
>
90
. Let the point
H
H
H
to be on
A
D
AD
A
D
such that
B
H
BH
B
H
is perpendicular to
A
D
AD
A
D
. Let the point
M
M
M
to be the midpoint of
A
B
AB
A
B
. Let the point
K
K
K
to be the intersecting point of the line
D
M
DM
D
M
with the circumcircle of
A
D
B
ADB
A
D
B
. Prove that
H
K
C
D
HKCD
HK
C
D
is concyclic.
1
2
Hide problems
Sequences, Sequences, Sequences
Consider the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
0
=
4
a_0=4
a
0
=
4
,
a
1
=
22
a_1=22
a
1
=
22
, and
a
n
−
6
a
n
−
1
+
a
n
−
2
=
0
a_n-6a_{n-1}+a_{n-2}=0
a
n
−
6
a
n
−
1
+
a
n
−
2
=
0
for
n
≥
2
n\ge2
n
≥
2
. Prove that there exist sequences
{
x
n
}
\{x_n\}
{
x
n
}
and
{
y
n
}
\{y_n\}
{
y
n
}
of positive integers such that
a
n
=
y
n
2
+
7
x
n
−
y
n
a_n=\frac{y_n^2+7}{x_n-y_n}
a
n
=
x
n
−
y
n
y
n
2
+
7
for any
n
≥
0
n\ge0
n
≥
0
.
Points of a grid with weights on them!
Let
n
≥
2
n \geq 2
n
≥
2
be a given integer. At any point
(
i
,
j
)
(i, j)
(
i
,
j
)
with
i
,
j
∈
Z
i, j \in\mathbb{ Z}
i
,
j
∈
Z
we write the remainder of
i
+
j
i+j
i
+
j
modulo
n
n
n
. Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers such that the rectangle with vertices
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
a
,
0
)
(a, 0)
(
a
,
0
)
,
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
0
,
b
)
(0, b)
(
0
,
b
)
has the following properties: (i) the remainders
0
,
1
,
…
,
n
−
1
0, 1, \ldots , n-1
0
,
1
,
…
,
n
−
1
written at its interior points appear the same number of times; (ii) the remainders
0
,
1
,
…
,
n
−
1
0, 1, \ldots , n -1
0
,
1
,
…
,
n
−
1
written at its boundary points appear the same number of times.
3
2
Hide problems
Bulgaria 2001
Given a permutation
(
a
1
,
a
1
,
.
.
.
,
a
n
)
(a_{1}, a_{1},...,a_{n})
(
a
1
,
a
1
,
...
,
a
n
)
of the numbers
1
,
2
,
.
.
.
,
n
1, 2,...,n
1
,
2
,
...
,
n
one may interchange any two consecutive "blocks" - that is, one may transform (
a
1
,
a
2
,
.
.
.
,
a
i
a_{1}, a_{2},...,a_{i}
a
1
,
a
2
,
...
,
a
i
,
a
i
+
1
,
.
.
.
a
i
+
p
,
⏟
A
\underbrace {a_{i+1},... a_{i+p},}_{A}
A
a
i
+
1
,
...
a
i
+
p
,
a
i
+
p
+
1
,
.
.
.
,
a
i
+
q
,
⏟
B
.
.
.
,
a
n
)
\underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n})
B
a
i
+
p
+
1
,
...
,
a
i
+
q
,
...
,
a
n
)
into
(
a
1
,
a
2
,
.
.
.
,
a
i
,
(a_{1}, a_{2},...,a_{i},
(
a
1
,
a
2
,
...
,
a
i
,
a
i
+
p
+
1
,
.
.
.
,
a
i
+
q
,
⏟
B
\underbrace {a_{i+p+1},...,a_{i+q},}_{B}
B
a
i
+
p
+
1
,
...
,
a
i
+
q
,
a
i
+
1
,
.
.
.
a
i
+
p
⏟
A
\underbrace {a_{i+1},... a_{i+p}}_{A}
A
a
i
+
1
,
...
a
i
+
p
,
.
.
.
,
a
n
)
,...,a_{n})
,
...
,
a
n
)
by interchanging the "blocks"
A
A
A
and
B
B
B
. Find the least number of such changes which are needed to transform
(
n
,
n
−
1
,
.
.
.
,
1
)
(n, n-1,...,1)
(
n
,
n
−
1
,
...
,
1
)
into
(
1
,
2
,
.
.
.
,
n
)
(1,2,...,n)
(
1
,
2
,
...
,
n
)
Prime Equation
Let
p
p
p
be a prime number congruent to
3
3
3
modulo
4
4
4
, and consider the equation
(
p
+
2
)
x
2
−
(
p
+
1
)
y
2
+
p
x
+
(
p
+
2
)
y
=
1
(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1
(
p
+
2
)
x
2
−
(
p
+
1
)
y
2
+
p
x
+
(
p
+
2
)
y
=
1
. Prove that this equation has infinitely many solutions in positive integers, and show that if
(
x
,
y
)
=
(
x
0
,
y
0
)
(x,y) = (x_{0}, y_{0})
(
x
,
y
)
=
(
x
0
,
y
0
)
is a solution of the equation in positive integers, then
p
∣
x
0
p | x_{0}
p
∣
x
0
.