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National and Regional Contests
Germany Contests
Germany Team Selection Test
2004 Germany Team Selection Test
2004 Germany Team Selection Test
Part of
Germany Team Selection Test
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(4)
4
1
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Looks like number theory but actually an inequality problem
Let the positive integers
x
1
x_1
x
1
,
x
2
x_2
x
2
,
.
.
.
...
...
,
x
100
x_{100}
x
100
satisfy the equation
1
x
1
+
1
x
2
+
.
.
.
+
1
x
100
=
20.
\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.
x
1
1
+
x
2
1
+
...
+
x
100
1
=
20.
Show that at least two of these integers are equal to each other.
1
5
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3
5
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2
5
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