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India National Olympiad
2015 India National Olympiad
6
Inmo 2015
Inmo 2015
Source: inmo 2015 #6
February 1, 2015
quadratics
modular arithmetic
pigeonhole principle
number theory
Problem Statement
Show that from a set of
11
11
11
square integers one can select six numbers
a
2
,
b
2
,
c
2
,
d
2
,
e
2
,
f
2
a^2,b^2,c^2,d^2,e^2,f^2
a
2
,
b
2
,
c
2
,
d
2
,
e
2
,
f
2
such that
a
2
+
b
2
+
c
2
≡
d
2
+
e
2
+
f
2
(
m
o
d
12
)
a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}
a
2
+
b
2
+
c
2
≡
d
2
+
e
2
+
f
2
(
mod
12
)
.
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