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Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2019 Nigeria Senior MO Round 2
2019 Nigeria Senior MO Round 2
Part of
Nigerian Senior Mathematics Olympiad Round 2
Subcontests
(6)
3
1
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3 externally tangent circles , PA is diameter of circle \Omega_a wanted
Circles
Ω
a
\Omega_a
Ω
a
and
Ω
b
\Omega_b
Ω
b
are externally tangent at
D
D
D
, circles
Ω
b
\Omega_b
Ω
b
and
Ω
c
\Omega_c
Ω
c
are externally tangent at
E
E
E
, circles
Ω
a
\Omega_a
Ω
a
and
Ω
c
\Omega_c
Ω
c
are externally tangent at
F
F
F
. Let
P
P
P
be an arbitrary point on
Ω
a
\Omega_a
Ω
a
different from
D
D
D
and
F
F
F
. Extend
P
D
PD
P
D
to meet
Ω
b
\Omega_b
Ω
b
again at
B
B
B
, extend
B
E
BE
BE
to meet
Ω
c
\Omega_c
Ω
c
again at
C
C
C
and extend
C
F
CF
CF
to meet
Ω
a
\Omega_a
Ω
a
again at
A
A
A
. Show that
P
A
PA
P
A
is a diameter of circle
Ω
a
\Omega_a
Ω
a
.
6
1
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Three square theorem
Let
N
=
4
K
L
N=4^KL
N
=
4
K
L
where
L
≡
7
(
m
o
d
8
)
L\equiv\ 7\pmod 8
L
≡
7
(
mod
8
)
. Prove that
N
N
N
cannot be written as a sum of 3 squares
5
1
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Murihead inequality
Let
a
a
a
,
b
b
b
, and
c
c
c
be real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. prove that
1
+
a
+
a
b
1
+
b
+
a
b
\frac{1+a+ab}{1+b+ab}
1
+
b
+
ab
1
+
a
+
ab
+
1
+
b
+
b
c
1
+
c
+
b
c
\frac{1+b+bc}{1+c+bc}
1
+
c
+
b
c
1
+
b
+
b
c
+
1
+
c
+
a
c
1
+
a
+
a
c
\frac{1+c+ac}{1+a+ac}
1
+
a
+
a
c
1
+
c
+
a
c
>
=
3
>=3
>=
3
4
1
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Polynomial gcd
Let
h
(
t
)
h(t)
h
(
t
)
and
f
(
t
)
f(t)
f
(
t
)
be polynomials such that
h
(
t
)
=
t
2
h(t)=t^2
h
(
t
)
=
t
2
and
f
n
(
t
)
=
h
(
h
(
h
(
h
(
h
.
.
.
h
(
t
)
)
)
)
)
)
−
1
f_n(t)=h(h(h(h(h...h(t))))))-1
f
n
(
t
)
=
h
(
h
(
h
(
h
(
h
...
h
(
t
))))))
−
1
where
h
(
t
)
h(t)
h
(
t
)
occurs
n
n
n
times. Prove that
f
n
(
t
)
f_n(t)
f
n
(
t
)
is a factor of
f
N
(
t
)
f_N(t)
f
N
(
t
)
whenever
n
n
n
is a factor of
N
N
N
2
1
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Easy divisibility
Suppose that
p
∣
(
2
t
2
−
1
)
p|(2t^2-1)
p
∣
(
2
t
2
−
1
)
and
p
2
∣
(
2
s
t
+
1
)
p^2|(2st+1)
p
2
∣
(
2
s
t
+
1
)
. Prove that
p
2
∣
(
s
2
+
t
2
−
1
)
p^2|(s^2+t^2-1)
p
2
∣
(
s
2
+
t
2
−
1
)
1
1
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Prove Fermat??
Prove that every prime of the form
4
k
+
1
4k+1
4
k
+
1
is the hypotenuse of a rectangular triangle with integer sides.