MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania - Local Contests
Unirea
2005 Unirea
2005 Unirea
Part of
Unirea
Subcontests
(2)
3
1
Hide problems
math contest 3
a
1
=
b
1
=
1
a_1=b_1=1
a
1
=
b
1
=
1
a
n
+
1
=
b
n
+
1
n
a_{n+1}=b_n+\frac{1}{n}
a
n
+
1
=
b
n
+
n
1
b
n
+
1
=
a
n
−
1
n
b_{n+1}=a_n-\frac{1}{n}
b
n
+
1
=
a
n
−
n
1
Prove that
a
n
a_n
a
n
,
b
n
b_n
b
n
is not convergent, but
a
n
b
n
a_nb_n
a
n
b
n
is convergent Laurentin Panaitopol
4
2
Hide problems
math constest 2
Find all
a
a
a
real number such that
x
n
=
n
{
a
n
!
}
x_n=n\{an! \}
x
n
=
n
{
an
!}
is convergeant Gabriel Dospinescu
Math constest 1
a
>
0
a>0
a
>
0
f
:
[
−
a
,
a
]
→
R
f:[-a,a]\rightarrow R
f
:
[
−
a
,
a
]
→
R
such that
f
′
′
f''
f
′′
exist and Riemann-integrable suppose
f
(
a
)
=
f
(
−
a
)
f(a)=f(-a)
f
(
a
)
=
f
(
−
a
)
f
′
(
−
a
)
=
f
′
(
a
)
=
a
2
f'(-a)=f'(a)=a^2
f
′
(
−
a
)
=
f
′
(
a
)
=
a
2
Prove that
6
a
3
≤
∫
−
a
a
f
′
′
(
x
)
2
d
x
6a^3\leq \int_{-a}^{a}{f''(x)}^2dx
6
a
3
≤
∫
−
a
a
f
′′
(
x
)
2
d
x
Study equality case ? Radu Miculescu