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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2003 Bulgaria National Olympiad
2003 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
2
2
Hide problems
An arbitrary line through O meets the sides of quad CNHM
Let
H
H
H
be an arbitrary point on the altitude
C
P
CP
CP
of the acute triangle
A
B
C
ABC
A
BC
. The lines
A
H
AH
A
H
and
B
H
BH
B
H
intersect
B
C
BC
BC
and
A
C
AC
A
C
in
M
M
M
and
N
N
N
, respectively.(a) Prove that
∠
N
P
C
=
∠
M
P
C
\angle NPC =\angle MPC
∠
NPC
=
∠
MPC
. (b) Let
O
O
O
be the common point of
M
N
MN
MN
and
C
P
CP
CP
. An arbitrary line through
O
O
O
meets the sides of quadrilateral
C
N
H
M
CNHM
CN
H
M
in
D
D
D
and
E
E
E
. Prove that
∠
E
P
C
=
∠
D
P
C
\angle EPC =\angle DPC
∠
EPC
=
∠
D
PC
.
abc can be written as a ratio of a cube and square
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be rational numbers such that
a
+
b
+
c
a+b+c
a
+
b
+
c
and
a
2
+
b
2
+
c
2
a^2+b^2+c^2
a
2
+
b
2
+
c
2
are equal integers. Prove that the number
a
b
c
abc
ab
c
can be written as the ratio of a perfect cube and a perfect square which are relatively prime.
1
2
Hide problems
Some n distinct sums of the form x_p+x_q+x_r are zero
Let
x
1
,
x
2
…
,
x
5
x_1, x_2 \ldots , x_5
x
1
,
x
2
…
,
x
5
be real numbers. Find the least positive integer
n
n
n
with the following property: if some
n
n
n
distinct sums of the form
x
p
+
x
q
+
x
r
x_p+x_q+x_r
x
p
+
x
q
+
x
r
(with
1
≤
p
<
q
<
r
≤
5
1\le p<q<r\le 5
1
≤
p
<
q
<
r
≤
5
) are equal to
0
0
0
, then
x
1
=
x
2
=
⋯
=
x
5
=
0
x_1=x_2=\cdots=x_5=0
x
1
=
x
2
=
⋯
=
x
5
=
0
.
Removing any element:Separated into 2 subsets with same sum
A set
A
A
A
of positive integers is called uniform if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer
n
>
1
n>1
n
>
1
such that there exist a uniform set
A
A
A
with
n
n
n
elements.
3
2
Hide problems
All the term of the sequence are perfect squares
Given the sequence
{
y
n
}
n
=
1
∞
\{y_n\}_{n=1}^{\infty}
{
y
n
}
n
=
1
∞
defined by
y
1
=
y
2
=
1
y_1=y_2=1
y
1
=
y
2
=
1
and
y
n
+
2
=
(
4
k
−
5
)
y
n
+
1
−
y
n
+
4
−
2
k
,
n
≥
1
y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1
y
n
+
2
=
(
4
k
−
5
)
y
n
+
1
−
y
n
+
4
−
2
k
,
n
≥
1
find all integers
k
k
k
such that every term of the sequence is a perfect square.
P(x)=2^n has an integer root
Determine all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that, for any positive integer
n
n
n
, the equation
P
(
x
)
=
2
n
P(x)=2^n
P
(
x
)
=
2
n
has an integer root.