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National and Regional Contests
Canada Contests
Canada National Olympiad
1974 Canada National Olympiad
1
Two identities
Two identities
Source: Canada 1974/1
January 13, 2007
logarithms
induction
Problem Statement
i) If
x
=
(
1
+
1
n
)
n
x = \left(1+\frac{1}{n}\right)^{n}
x
=
(
1
+
n
1
)
n
and
y
=
(
1
+
1
n
)
n
+
1
y=\left(1+\frac{1}{n}\right)^{n+1}
y
=
(
1
+
n
1
)
n
+
1
, show that
y
x
=
x
y
y^{x}= x^{y}
y
x
=
x
y
. ii) Show that, for all positive integers
n
n
n
,
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
−
1
)
n
(
n
−
1
)
2
+
(
−
1
)
n
+
1
n
2
=
(
−
1
)
n
+
1
(
1
+
2
+
⋯
+
n
)
.
1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
−
1
)
n
(
n
−
1
)
2
+
(
−
1
)
n
+
1
n
2
=
(
−
1
)
n
+
1
(
1
+
2
+
⋯
+
n
)
.
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