MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania - Local Contests
Gheorghe Vranceanu
2010 Gheorghe Vranceanu
2010 Gheorghe Vranceanu
Part of
Gheorghe Vranceanu
Subcontests
(4)
4
1
Hide problems
purely NT problem with sequences not necessarily composed of numbers
Let be two real numbers
α
,
β
\alpha ,\beta
α
,
β
and two sequences
(
x
n
)
n
≥
1
,
(
y
n
)
n
≥
1
\left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1}
(
x
n
)
n
≥
1
,
(
y
n
)
n
≥
1
whose smallest periods are
p
,
q
,
p,q,
p
,
q
,
respectively. Prove that the sequence
(
α
x
n
+
β
y
n
)
n
≥
1
\left( \alpha x_n+\beta y_n\right)_{n\ge 1}
(
α
x
n
+
β
y
n
)
n
≥
1
is periodic if
gcd
2
(
p
,
q
)
∣
lcm
(
p
,
q
)
,
\text{gcd}^2 (p,q) | \text{lcm} (p,q) ,
gcd
2
(
p
,
q
)
∣
lcm
(
p
,
q
)
,
and in this case find its smallest period.
3
1
Hide problems
kind of Goldbach, something to think
Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.
2
5
Show problems
1
5
Show problems