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Belarus Contests
Belarus Team Selection Test
2009 Belarus Team Selection Test
2009 Belarus Team Selection Test
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Belarus Team Selection Test
Subcontests
(4)
4
2
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xy is perfect square if yx^2+(y^2-z^2)x+y(y-z)^2=0 in integers x,y,z
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be integer numbers satisfying the equality
y
x
2
+
(
y
2
−
z
2
)
x
+
y
(
y
−
z
)
2
=
0
yx^2+(y^2-z^2)x+y(y-z)^2=0
y
x
2
+
(
y
2
−
z
2
)
x
+
y
(
y
−
z
)
2
=
0
a) Prove that number
x
y
xy
x
y
is a perfect square. b) Prove that there are infinitely many triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
satisfying the equality.I.Voronovich
min no of graphs such any 2 vertices are connected with a third vertex by edges
Given a graph with
n
n
n
(
n
≥
4
n\ge 4
n
≥
4
) vertices . It is known that for any two vertices
A
A
A
and
B
B
B
there exists a vertex which is connected by edges both with
A
A
A
and
B
B
B
. Find the smallest possible numbers of edges in the graph. E. Barabanov
3
4
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2
3
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find n such that x^n+y^n+z^n is constant for all x,y,z x+y+z=0 and xyz=1.
Find all
n
∈
N
n \in N
n
∈
N
for which the value of the expression
x
n
+
y
n
+
z
n
x^n+y^n+z^n
x
n
+
y
n
+
z
n
is constant for all
x
,
y
,
z
∈
R
x,y,z \in R
x
,
y
,
z
∈
R
such that
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
and
x
y
z
=
1
xyz=1
x
yz
=
1
.D. Bazylev
x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}} cannot an be infitine sequence
a) Prove that there is not an infinte sequence
(
x
n
)
(x_n)
(
x
n
)
,
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
of positive real numbers satisfying the relation
x
n
+
2
=
x
n
+
1
−
x
n
x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}
x
n
+
2
=
x
n
+
1
−
x
n
,
∀
n
∈
N
\forall n \in N
∀
n
∈
N
(*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms? I.Voronovich
XA_i=A_{i+2}A_{i+3} inside a convex pentagon
Does there exist a convex pentagon
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
and a point
X
X
X
inside it such that
X
A
i
=
A
i
+
2
A
i
+
3
XA_i=A_{i+2}A_{i+3}
X
A
i
=
A
i
+
2
A
i
+
3
for all
i
=
1
,
.
.
.
,
5
i=1,...,5
i
=
1
,
...
,
5
(all indices are considered modulo
5
5
5
) ?I. Voronovich
1
8
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