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International Contests
Tournament Of Towns
1991 Tournament Of Towns
(302) 3
TOT 302 1991 Autumn O J3 nested fractions sum
TOT 302 1991 Autumn O J3 nested fractions sum
Source:
June 9, 2024
algebra
Sum
Problem Statement
Prove that
1
2
+
1
3
+
1
4
+
1
.
.
.
+
1
9991
+
1
1
+
1
1
+
1
3
+
1
4
+
1
.
.
.
+
1
9991
=
1
\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1
2
+
3
+
4
+
...
+
9991
1
1
1
1
1
+
1
+
1
+
3
+
4
+
...
+
9991
1
1
1
1
1
1
=
1
This means
1
/
(
2
+
(
1
/
(
3
+
(
1
/
(
4
+
(
.
.
.
+
1
/
1991
)
)
)
)
)
)
+
1
/
(
1
+
(
1
/
(
1
+
(
1
/
(
3
+
(
1
/
(
4
+
(
.
.
.
+
1
/
1991...
)
)
)
)
)
)
)
)
=
1.
)
1/(2+ (1/(3+ (1/(4+(...+1/1991)))))) +1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)
1/
(
2
+
(
1/
(
3
+
(
1/
(
4
+
(
...
+
1/1991
))))))
+
1/
(
1
+
(
1/
(
1
+
(
1/
(
3
+
(
1/
(
4
+
(
...
+
1/1991...
))))))))
=
1.
)
(G. Galperin, Moscow-Tel Aviv)
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