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2012 Bosnia Herzegovina Team Selection Test
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Part of
2012 Bosnia Herzegovina Team Selection Test
Problems
(1)
Bosnia and Herzegovina TST 2012 Problem 3
Source:
5/19/2012
Prove that for all odd prime numbers
p
p
p
there exist a natural number
m
<
p
m<p
m
<
p
and integers
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
such that:
m
p
=
x
1
2
+
x
2
2
+
x
3
2
.
mp=x_1^2+x_2^2+x_3^2.
m
p
=
x
1
2
+
x
2
2
+
x
3
2
.
calculus
integration
modular arithmetic
quadratics
number theory
prime numbers
number theory proposed