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Putnam
1978 Putnam
B4
B4
Part of
1978 Putnam
Problems
(1)
Putnam 1978 B4
Source: Putnam 1978
5/3/2022
Prove that for every real number
N
N
N
the equation
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
=
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
2
x
3
x
4
x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
=
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
2
x
3
x
4
has an integer solution
(
x
1
,
x
2
,
x
3
,
x
4
)
(x_1 , x_2 , x_3 , x_4)
(
x
1
,
x
2
,
x
3
,
x
4
)
for which
x
1
,
x
2
,
x
3
x_1, x_2 , x_3
x
1
,
x
2
,
x
3
and
x
4
x_4
x
4
are all larger than
N
.
N.
N
.
Putnam
Integers
equation