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2016 Latvia National Olympiad 3rd Round Grade11Problem3
2016 Latvia National Olympiad 3rd Round Grade11Problem3
Source:
July 22, 2016
algebra
equation
egyptian fractions
Problem Statement
Prove that for every integer
n
n
n
(
n
>
1
n > 1
n
>
1
) there exist two positive integers
x
x
x
and
y
y
y
(
x
≤
y
x \leq y
x
≤
y
) such that
1
n
=
1
x
(
x
+
1
)
+
1
(
x
+
1
)
(
x
+
2
)
+
⋯
+
1
y
(
y
+
1
)
\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}
n
1
=
x
(
x
+
1
)
1
+
(
x
+
1
)
(
x
+
2
)
1
+
⋯
+
y
(
y
+
1
)
1
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