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National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
1997 Switzerland Team Selection Test
1
1
Part of
1997 Switzerland Team Selection Test
Problems
(1)
Swiss tst
Source: Swiss IMO Team Selection Test 1997
5/7/2017
1. A finite sequence of integers
a
0
,
a
1
,
.
.
.
,
a
n
a_0,a_1,...,a_n
a
0
,
a
1
,
...
,
a
n
is called quadratic if
∣
a
k
−
a
k
−
1
∣
=
k
2
|a_k -a_{k-1}| = k^2
∣
a
k
−
a
k
−
1
∣
=
k
2
for
n
≥
k
≥
1
n\geq k\geq1
n
≥
k
≥
1
. (a) Prove that for any two integers
b
b
b
and
c
c
c
, there exist a natural number
n
n
n
and a quadratic sequence with
a
0
=
b
a_0 = b
a
0
=
b
and
a
n
=
c
a_n =c
a
n
=
c
. (b) Find the smallest natural number
n
n
n
for which there exists a quadratic sequence with
a
0
=
0
a_0 = 0
a
0
=
0
and
a
n
=
1997
a_n = 1997
a
n
=
1997
algebra
Sequence