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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1949 Moscow Mathematical Olympiad
158
158
Part of
1949 Moscow Mathematical Olympiad
Problems
(1)
MMO 158 Moscow MO 1949 diophantine x^2 + y^2 + z^2 + u^2 = 2xyzu
Source:
7/20/2019
a) Prove that
x
2
+
y
2
+
z
2
=
2
x
y
z
x^2 + y^2 + z^2 = 2xyz
x
2
+
y
2
+
z
2
=
2
x
yz
for integer
x
,
y
,
z
x, y, z
x
,
y
,
z
only if
x
=
y
=
z
=
0
x = y = z = 0
x
=
y
=
z
=
0
.b) Find integers
x
,
y
,
z
,
u
x, y, z, u
x
,
y
,
z
,
u
such that
x
2
+
y
2
+
z
2
+
u
2
=
2
x
y
z
u
x^2 + y^2 + z^2 + u^2 = 2xyzu
x
2
+
y
2
+
z
2
+
u
2
=
2
x
yz
u
.
diophantine
number theory