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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2020 Nigerian Senior MO Round 2
2020 Nigerian Senior MO Round 2
Part of
Nigerian Senior Mathematics Olympiad Round 2
Subcontests
(5)
3
1
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Lines and intersection
N
N
N
straight lines are drawn on a plane. The
N
N
N
lines can be partitioned into set of lines such that if a line
l
l
l
belongs to a partition set then all lines parallel to
l
l
l
make up the rest of that set. For each
n
>
=
1
n>=1
n
>=
1
,let
a
n
a_n
a
n
denote the number of partition sets of size
n
n
n
. Now that
N
N
N
lines intersect at certain points on the plane. For each
n
>
=
2
n>=2
n
>=
2
let
b
n
b_n
b
n
denote the number of points that are intersection of exactly
n
n
n
lines. Show that
∑
n
>
=
2
(
a
n
+
b
n
)
\sum_{n>= 2}(a_n+b_n)
∑
n
>=
2
(
a
n
+
b
n
)
(
n
2
)
\binom{n}{2}
(
2
n
)
=
=
=
(
N
2
)
\binom{N}{2}
(
2
N
)
5
1
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Maximum value
Given that
f
(
x
)
=
(
3
+
2
x
)
3
(
4
−
x
)
4
f(x)=(3+2x)^3(4-x)^4
f
(
x
)
=
(
3
+
2
x
)
3
(
4
−
x
)
4
on the interval
−
3
2
<
x
<
4
\frac{-3}{2}<x<4
2
−
3
<
x
<
4
. Find the a. Maximum value of
f
(
x
)
f(x)
f
(
x
)
b. The value of
x
x
x
that gives the maximum in (a).
4
1
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Primes squares odd condition
Let
N
>
=
2
N>= 2
N
>=
2
be an integer. Show that
4
n
(
N
−
n
)
+
1
4n(N-n)+1
4
n
(
N
−
n
)
+
1
is never a perfect square for each natural number
n
n
n
less than
N
N
N
if and only if
N
2
+
1
N^2+1
N
2
+
1
is prime.
2
1
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Menelaus theorem
Let
D
D
D
be a point in the interior of
A
B
C
ABC
A
BC
. Let
B
D
BD
B
D
and
A
C
AC
A
C
intersect at
E
E
E
while
C
D
CD
C
D
and
A
B
AB
A
B
intersect at
F
F
F
. Let
E
F
EF
EF
intersect
B
C
BC
BC
at
G
G
G
. Let
H
H
H
be an arbitrary point on
A
D
AD
A
D
. Let
H
F
HF
H
F
and
B
D
BD
B
D
intersect at
I
I
I
. Let
H
E
HE
H
E
and
C
D
CD
C
D
intersect at
J
J
J
. prove that
G
G
G
,
I
I
I
and
J
J
J
are collinear
1
1
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Viete and polynomials
Let
k
k
k
be a real number. Define on the set of reals the operation
x
∗
y
x*y
x
∗
y
=
x
y
x
+
y
+
k
\frac{xy}{x+y+k}
x
+
y
+
k
x
y
whenever
x
+
y
x+y
x
+
y
does not equal
−
k
-k
−
k
. Let
x
1
<
x
2
<
x
3
<
x
4
x_1<x_2<x_3<x_4
x
1
<
x
2
<
x
3
<
x
4
be the roots of
t
4
=
27
(
t
2
+
t
+
1
)
t^4=27(t^2+t+1)
t
4
=
27
(
t
2
+
t
+
1
)
.suppose that
[
(
x
1
∗
x
2
)
∗
x
3
]
∗
x
4
=
1
[(x_1*x_2)*x_3]*x_4=1
[(
x
1
∗
x
2
)
∗
x
3
]
∗
x
4
=
1
. Find all possible values of
k
k
k