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Bogdan Stan
2013 Bogdan Stan
2013 Bogdan Stan
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Bogdan Stan
Subcontests
(4)
4
3
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logarithmic equation
Solve in the real numbers the equation
3
x
−
1
3
(
1
−
log
3
3
x
)
=
1.
3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1.
3
3
x
−
1
(
1
−
lo
g
3
3
x
)
=
1.
Ion Gușatu
Nice pigeonhole problem
Consider
16
16
16
pairwise distinct natural numbers smaller than
1597.
1597.
1597.
a) Prove that among these, there are three numbers having the property that the sum of any two of them is bigger than the third. b) If one of these numbers is
1597
,
1597,
1597
,
is still true the fact from subpoint a)? Teodor Radu
Nice application of Cesaro-Stolz
Let be a sequence
(
x
n
)
n
≥
1
\left( x_n \right)_{n\ge 1}
(
x
n
)
n
≥
1
having the property that
lim
n
→
∞
(
14
(
n
+
2
)
x
n
+
2
−
15
(
n
+
1
)
x
n
+
1
+
n
x
n
)
=
13.
\lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13.
n
→
∞
lim
(
14
(
n
+
2
)
x
n
+
2
−
15
(
n
+
1
)
x
n
+
1
+
n
x
n
)
=
13.
Show that
(
x
n
)
n
≥
1
\left( x_n \right)_{n\ge 1}
(
x
n
)
n
≥
1
is convergent and calculate its limit. Cosmin Nițu
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4
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