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France Contests
French Mathematical Olympiad
1990 French Mathematical Olympiad
Problem 3
Problem 3
Part of
1990 French Mathematical Olympiad
Problems
(1)
diophantine equation involving reciprocals of squares
Source: France 1990 P3
5/18/2021
(a) Find all triples of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
for which
1
4
=
1
a
2
+
1
b
2
+
1
c
2
\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}
4
1
=
a
2
1
+
b
2
1
+
c
2
1
. (b) Determine all positive integers
n
n
n
for which there exist positive integers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
such that
1
=
1
x
1
2
+
1
x
2
2
+
…
+
1
x
n
2
1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}
1
=
x
1
2
1
+
x
2
2
1
+
…
+
x
n
2
1
.
number theory
Diophantine equation