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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2001 Junior Balkan Team Selection Tests - Moldova
6
6
Part of
2001 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
2 <= i <= 8 there is a number $a_i$, such that a_{i - 1} + a_{i + 1} <ka_i
Source: 2001 Moldova JBMO TST p6
2/20/2021
Let the nonnegative numbers
a
1
,
a
2
,
.
.
.
a
9
a_1, a_2,... a_9
a
1
,
a
2
,
...
a
9
, where
a
1
=
a
9
=
0
a_1 = a_9 = 0
a
1
=
a
9
=
0
and let at least one of the numbers is nonzero. Denote the sentence
(
P
)
(P)
(
P
)
: '' For
2
≤
i
≤
8
2 \le i \le 8
2
≤
i
≤
8
there is a number
a
i
a_i
a
i
, such that
a
i
−
1
+
a
i
+
1
<
k
a
i
a_{i - 1} + a_{i + 1} <ka_i
a
i
−
1
+
a
i
+
1
<
k
a
i
”. a) Show that the sentence
(
P
)
(P)
(
P
)
is true for
k
=
2
k = 2
k
=
2
. b) Determine whether is the sentence
(
P
)
(P)
(
P
)
true for
k
=
19
10
k = \frac{19}{10}
k
=
10
19
inequalities
algebra