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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2004 Bulgaria National Olympiad
2004 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
6
1
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so much nt in bulgarian mo 2004 :) [Cauchy-Davenport theorem
Let
p
p
p
be a prime number and let
0
≤
a
1
<
a
2
<
⋯
<
a
m
<
p
0\leq a_{1}< a_{2}<\cdots < a_{m}< p
0
≤
a
1
<
a
2
<
⋯
<
a
m
<
p
and
0
≤
b
1
<
b
2
<
⋯
<
b
n
<
p
0\leq b_{1}< b_{2}<\cdots < b_{n}< p
0
≤
b
1
<
b
2
<
⋯
<
b
n
<
p
be arbitrary integers. Let
k
k
k
be the number of distinct residues modulo
p
p
p
that a_{i}\plus{}b_{j} give when
i
i
i
runs from 1 to
m
m
m
, and
j
j
j
from 1 to
n
n
n
. Prove that a) if m\plus{}n > p then k \equal{} p; b) if m\plus{}n\leq p then k\geq m\plus{}n\minus{}1.
5
1
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greatest common divisor
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive integers such that the number of pairs
(
x
,
y
)
∈
(
0
,
1
)
2
(x,y) \in (0,1)^2
(
x
,
y
)
∈
(
0
,
1
)
2
such that both
a
x
+
b
y
ax+by
a
x
+
b
y
and
c
x
+
d
y
cx+dy
c
x
+
d
y
are integers is equal with 2004. If
gcd
(
a
,
c
)
=
6
\gcd (a,c)=6
g
cd
(
a
,
c
)
=
6
find
gcd
(
b
,
d
)
\gcd (b,d)
g
cd
(
b
,
d
)
.
4
1
Hide problems
words
In a word formed with the letters
a
,
b
a,b
a
,
b
we can change some blocks:
a
b
a
aba
aba
in
b
b
b
and back,
b
b
a
bba
bba
in
a
a
a
and backwards. If the initial word is
a
a
a
…
a
b
aaa\ldots ab
aaa
…
ab
where
a
a
a
appears 2003 times can we reach the word
b
a
a
a
…
a
baaa\ldots a
baaa
…
a
, where
a
a
a
appears 2003 times.
3
1
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tourists
A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most
2
5
n
\displaystyle \frac 2{5}n
5
2
n
tourists.
2
1
Hide problems
looks like Fedor's russian problem
For any positive integer
n
n
n
the sum
1
+
1
2
+
⋯
+
1
n
\displaystyle 1+\frac 12+ \cdots + \frac 1n
1
+
2
1
+
⋯
+
n
1
is written in the form
P
(
n
)
Q
(
n
)
\displaystyle \frac{P(n)}{Q(n)}
Q
(
n
)
P
(
n
)
, where
P
(
n
)
P(n)
P
(
n
)
and
Q
(
n
)
Q(n)
Q
(
n
)
are relatively prime. a) Prove that
P
(
67
)
P(67)
P
(
67
)
is not divisible by 3; b) Find all possible
n
n
n
, for which
P
(
n
)
P(n)
P
(
n
)
is divisible by 3.
1
1
Hide problems
another incenter problem
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
, and let
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
be arbitrary points on the segments
(
A
I
)
(AI)
(
A
I
)
,
(
B
I
)
(BI)
(
B
I
)
,
(
C
I
)
(CI)
(
C
I
)
, respectively. The perpendicular bisectors of
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
intersect each other at
A
2
A_2
A
2
,
B
2
B_2
B
2
, and
C
2
C_2
C
2
. Prove that the circumcenter of the triangle
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
coincides with the circumcenter of the triangle
A
B
C
ABC
A
BC
if and only if
I
I
I
is the orthocenter of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
.