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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2012 Bulgaria National Olympiad
2012 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
3
2
Hide problems
Winning strategy involving coefficents of a polynomial
We are given a real number
a
a
a
, not equal to
0
0
0
or
1
1
1
. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
∗
x
4
+
∗
x
3
+
∗
x
2
+
∗
x
1
+
∗
=
0
*x^4+*x^3+*x^2+*x^1+*=0
∗
x
4
+
∗
x
3
+
∗
x
2
+
∗
x
1
+
∗
=
0
with a number of the type
a
n
a^n
a
n
, where
n
n
n
is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of
a
a
a
) who has a winning strategy
Circumcircle of three points passes through fixed point
We are given an acute-angled triangle
A
B
C
ABC
A
BC
and a random point
X
X
X
in its interior, different from the centre of the circumcircle
k
k
k
of the triangle. The lines
A
X
,
B
X
AX,BX
A
X
,
BX
and
C
X
CX
CX
intersect
k
k
k
for a second time in the points
A
1
,
B
1
A_1,B_1
A
1
,
B
1
and
C
1
C_1
C
1
respectively. Let
A
2
,
B
2
A_2,B_2
A
2
,
B
2
and
C
2
C_2
C
2
be the points that are symmetric of
A
1
,
B
1
A_1,B_1
A
1
,
B
1
and
C
1
C_1
C
1
in respect to
B
C
,
A
C
BC,AC
BC
,
A
C
and
A
B
AB
A
B
respectively. Prove that the circumcircle of the triangle
A
2
,
B
2
A_2,B_2
A
2
,
B
2
and
C
2
C_2
C
2
passes through a constant point that does not depend on the choice of
X
X
X
.
2
2
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Two conditions for colouring natural numbers
Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number
p
p
p
and every natural number
n
n
n
, the numbers
p
n
,
p
n
+
1
p^n,p^{n+1}
p
n
,
p
n
+
1
and
p
n
+
2
p^{n+2}
p
n
+
2
do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.
P(x) + P(y) + P(z) > 0
Let
Q
(
x
)
Q(x)
Q
(
x
)
be a quadratic trinomial. Given that the function
P
(
x
)
=
x
2
Q
(
x
)
P(x)=x^{2}Q(x)
P
(
x
)
=
x
2
Q
(
x
)
is increasing in the interval
(
0
,
∞
)
(0,\infty )
(
0
,
∞
)
, prove that:
P
(
x
)
+
P
(
y
)
+
P
(
z
)
>
0
P(x) + P(y) + P(z) > 0
P
(
x
)
+
P
(
y
)
+
P
(
z
)
>
0
for all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
+
y
+
z
>
0
x+y+z>0
x
+
y
+
z
>
0
and
x
y
z
>
0
xyz>0
x
yz
>
0
.
1
2
Hide problems
Two consecutive terms in the sequence are squares
The sequence
a
1
,
a
2
,
a
3
…
a_1,a_2,a_3\ldots
a
1
,
a
2
,
a
3
…
, consisting of natural numbers, is defined by the rule:
a
n
+
1
=
a
n
+
2
t
(
n
)
a_{n+1}=a_{n}+2t(n)
a
n
+
1
=
a
n
+
2
t
(
n
)
for every natural number
n
n
n
, where
t
(
n
)
t(n)
t
(
n
)
is the number of the different divisors of
n
n
n
(including
1
1
1
and
n
n
n
). Is it possible that two consecutive members of the sequence are squares of natural numbers?
Partitioning the set of binary sequence with a condition
Let
n
n
n
be an even natural number and let
A
A
A
be the set of all non-zero sequences of length
n
n
n
, consisting of numbers
0
0
0
and
1
1
1
(length
n
n
n
binary sequences, except the zero sequence
(
0
,
0
,
…
,
0
)
(0,0,\ldots,0)
(
0
,
0
,
…
,
0
)
). Prove that
A
A
A
can be partitioned into groups of three elements, so that for every triad
{
(
a
1
,
a
2
,
…
,
a
n
)
,
(
b
1
,
b
2
,
…
,
b
n
)
,
(
c
1
,
c
2
,
…
,
c
n
)
}
\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}
{(
a
1
,
a
2
,
…
,
a
n
)
,
(
b
1
,
b
2
,
…
,
b
n
)
,
(
c
1
,
c
2
,
…
,
c
n
)}
, and for every
i
=
1
,
2
,
…
,
n
i = 1, 2,\ldots,n
i
=
1
,
2
,
…
,
n
, exactly zero or two of the numbers
a
i
,
b
i
,
c
i
a_i, b_i, c_i
a
i
,
b
i
,
c
i
are equal to
1
1
1
.