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2013 IMO Shortlist
A1
IMO Shortlist 2013, Algebra #1
IMO Shortlist 2013, Algebra #1
Source: IMO Shortlist 2013, Algebra #1
July 9, 2014
algebra
Sequence
IMO Shortlist
Problem Statement
Let
n
n
n
be a positive integer and let
a
1
,
…
,
a
n
−
1
a_1, \ldots, a_{n-1}
a
1
,
…
,
a
n
−
1
be arbitrary real numbers. Define the sequences
u
0
,
…
,
u
n
u_0, \ldots, u_n
u
0
,
…
,
u
n
and
v
0
,
…
,
v
n
v_0, \ldots, v_n
v
0
,
…
,
v
n
inductively by
u
0
=
u
1
=
v
0
=
v
1
=
1
u_0 = u_1 = v_0 = v_1 = 1
u
0
=
u
1
=
v
0
=
v
1
=
1
, and
u
k
+
1
=
u
k
+
a
k
u
k
−
1
u_{k+1} = u_k + a_k u_{k-1}
u
k
+
1
=
u
k
+
a
k
u
k
−
1
,
v
k
+
1
=
v
k
+
a
n
−
k
v
k
−
1
v_{k+1} = v_k + a_{n-k} v_{k-1}
v
k
+
1
=
v
k
+
a
n
−
k
v
k
−
1
for
k
=
1
,
…
,
n
−
1.
k=1, \ldots, n-1.
k
=
1
,
…
,
n
−
1.
Prove that
u
n
=
v
n
.
u_n = v_n.
u
n
=
v
n
.
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