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1988 IMO Longlists
41
41
Part of
1988 IMO Longlists
Problems
(1)
Master of fractions for the new century wanted
Source: IMO LongList 1988, India 6, Problem 41 of ILL
11/3/2005
i.) Calculate
x
x
x
if
x
=
(
11
+
6
⋅
2
)
⋅
11
−
6
⋅
2
−
(
11
−
6
⋅
2
)
⋅
11
+
6
⋅
2
(
5
+
2
+
5
−
2
)
−
(
5
+
1
)
x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})}
x
=
(
5
+
2
+
5
−
2
)
−
(
5
+
1
)
(
11
+
6
⋅
2
)
⋅
11
−
6
⋅
2
−
(
11
−
6
⋅
2
)
⋅
11
+
6
⋅
2
ii.) For each positive number
x
,
x,
x
,
let
k
=
(
x
+
1
x
)
6
−
(
x
6
+
1
x
6
)
−
2
(
x
+
1
x
)
3
−
(
x
3
+
1
x
3
)
k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)}
k
=
(
x
+
x
1
)
3
−
(
x
3
+
x
3
1
)
(
x
+
x
1
)
6
−
(
x
6
+
x
6
1
)
−
2
Calculate the minimum value of
k
.
k.
k
.
algebra unsolved
algebra