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National and Regional Contests
India Contests
NMTC
2017 NMTC Junior
1
NT and algebra
NT and algebra
Source: NMTC 2017 Junior P1
October 28, 2017
algebra
number theory
Problem Statement
(a) Find all prime numbers
p
p
p
such that
4
p
2
+
1
4p^2+1
4
p
2
+
1
and
6
p
2
+
1
6p^2+1
6
p
2
+
1
are also primes.(b)Find real numbers
x
,
y
,
z
,
u
x,y,z,u
x
,
y
,
z
,
u
such that
x
y
z
+
x
y
+
y
z
+
z
x
+
x
+
y
+
z
=
7
xyz+xy+yz+zx+x+y+z=7
x
yz
+
x
y
+
yz
+
z
x
+
x
+
y
+
z
=
7
y
z
u
+
y
z
+
z
u
+
u
y
+
y
+
z
+
u
=
10
yzu+yz+zu+uy+y+z+u=10
yz
u
+
yz
+
z
u
+
u
y
+
y
+
z
+
u
=
10
z
u
x
+
z
u
+
u
x
+
x
z
+
z
+
u
+
x
=
10
zux+zu+ux+xz+z+u+x=10
z
ux
+
z
u
+
ux
+
x
z
+
z
+
u
+
x
=
10
u
x
y
+
u
x
+
x
y
+
y
u
+
u
+
x
+
y
=
10
uxy+ux+xy+yu+u+x+y=10
ux
y
+
ux
+
x
y
+
y
u
+
u
+
x
+
y
=
10
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