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Middle European Mathematical Olympiad
2020 Middle European Mathematical Olympiad
4#
finding n such that 1 is a sum of fractions
finding n such that 1 is a sum of fractions
Source: 2020 MEMO I-4
August 30, 2020
algebra
memo
MEMO 2020
equation
induction
Problem Statement
Find all positive integers
n
n
n
for which there exist positive integers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \dots, x_n
x
1
,
x
2
,
…
,
x
n
such that
1
x
1
2
+
2
x
2
2
+
2
2
x
3
2
+
⋯
+
2
n
−
1
x
n
2
=
1.
\frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.
x
1
2
1
+
x
2
2
2
+
x
3
2
2
2
+
⋯
+
x
n
2
2
n
−
1
=
1.
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