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National and Regional Contests
Romania Contests
Romania - Local Contests
SNSB Admissions
2002 SNSB Admission
2002 SNSB Admission
Part of
SNSB Admissions
Subcontests
(6)
6
1
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Galois extension of Q of type Z_3
Find a Galois extension of the field
Q
\mathbb{Q}
Q
whose Galois group is isomorphic with
Z
/
3
Z
.
\mathbb{Z}/3\mathbb{Z} .
Z
/3
Z
.
5
1
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A complex condition for a function to be 0
Let
f
:
D
⟶
C
f:\mathbb{D}\longrightarrow\mathbb{C}
f
:
D
⟶
C
be a continuous function, where
D
\mathbb{D}
D
is the closed unit disk. Suppose that
f
f
f
is holomorphic on the open unit disk and that
e
i
θ
e^{i\theta }
e
i
θ
are roots, for any
θ
∈
[
0
,
π
/
4
]
.
\theta\in\left[ 0,\pi /4 \right] .
θ
∈
[
0
,
π
/4
]
.
Show that
f
=
0
D
.
f=0_{\mathbb{D}} .
f
=
0
D
.
4
1
Hide problems
Pathological planar shapes of any measure
Present a family of subsets of the plane such that each one of its members is Lebesgue measurable, each one of its members intersects any circle, and the set of Lebesgue measures of all its members is the set of nonnegative real numbers.
3
1
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Classification of the topologies of the support of x |-> x²+axy+by²+cx+dy+e
Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.
2
1
Hide problems
Locally smooth solutions for monic polynomial equations
Provided that the roots of the polynom
X
n
+
a
1
X
n
−
1
+
a
2
X
n
−
2
+
⋯
+
a
n
−
1
X
+
a
n
:
∈
R
[
X
]
,
X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] ,
X
n
+
a
1
X
n
−
1
+
a
2
X
n
−
2
+
⋯
+
a
n
−
1
X
+
a
n
:∈
R
[
X
]
,
of degree
n
≥
2
,
n\ge 2,
n
≥
2
,
are all real and pairwise distinct, prove that there exists is a neighbourhood
V
\mathcal{V}
V
of
(
a
1
,
a
2
,
…
,
a
n
)
\left( a_1,a_2,\ldots ,a_n \right)
(
a
1
,
a
2
,
…
,
a
n
)
in
R
n
\mathbb{R}^n
R
n
and
n
n
n
functions
x
1
,
x
2
,
…
,
x
n
∈
C
∞
(
V
)
x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right)
x
1
,
x
2
,
…
,
x
n
∈
C
∞
(
V
)
whose values at
(
a
1
,
a
2
,
…
,
a
n
)
\left( a_1,a_2,\ldots ,a_n \right)
(
a
1
,
a
2
,
…
,
a
n
)
are roots of the mentioned polynom.
1
1
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Condition for the endomorphisms of finite vec. spaces to be nilpotent
Let
u
,
v
u,v
u
,
v
be two endomorphisms of a finite vectorial space that verify the relation
u
v
−
v
u
=
u
.
uv-vu=u.
uv
−
vu
=
u
.
Calculate
u
k
v
−
v
u
k
u^kv-vu^k
u
k
v
−
v
u
k
and show that u is nilpotent.