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1984 IMO Shortlist
19
Find the harmonic mean of the 1985th row
Find the harmonic mean of the 1985th row
Source:
September 8, 2010
algebra
combinatorics
recurrence relation
Linear Recurrences
IMO Shortlist
Problem Statement
The harmonic table is a triangular array:
1
1
1
1
2
1
2
\frac 12 \qquad \frac 12
2
1
2
1
1
3
1
6
1
3
\frac 13 \qquad \frac 16 \qquad \frac 13
3
1
6
1
3
1
1
4
1
12
1
12
1
4
\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14
4
1
12
1
12
1
4
1
Where
a
n
,
1
=
1
n
a_{n,1} = \frac 1n
a
n
,
1
=
n
1
and
a
n
,
k
+
1
=
a
n
−
1
,
k
−
a
n
,
k
a_{n,k+1} = a_{n-1,k} - a_{n,k}
a
n
,
k
+
1
=
a
n
−
1
,
k
−
a
n
,
k
for
1
≤
k
≤
n
−
1.
1 \leq k \leq n-1.
1
≤
k
≤
n
−
1.
Find the harmonic mean of the
198
5
t
h
1985^{th}
198
5
t
h
row.
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