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Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2024 JBMO TST - Turkey
6
6
Part of
2024 JBMO TST - Turkey
Problems
(1)
Interesting sequence for Juniors
Source: Turkey JBMO TST 2024 P6
5/13/2024
Let
(
a
n
)
n
=
0
∞
{(a_n)}_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
and
(
b
n
)
n
=
0
∞
{(b_n)}_{n=0}^{\infty}
(
b
n
)
n
=
0
∞
be real squences such that
a
0
=
40
a_0=40
a
0
=
40
,
b
0
=
41
b_0=41
b
0
=
41
and for all
n
≥
0
n\geq 0
n
≥
0
the given equalities hold.
a
n
+
1
=
a
n
+
1
b
n
and
b
n
+
1
=
b
n
+
1
a
n
a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}
a
n
+
1
=
a
n
+
b
n
1
and
b
n
+
1
=
b
n
+
a
n
1
Find the least possible positive integer value of
k
k
k
such that the value of
a
k
a_k
a
k
is strictly bigger than
80
80
80
.
algebra
Sequence