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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 3
2020 Nigerian MO round 3
2020 Nigerian MO round 3
Part of
Nigerian Senior Mathematics Olympiad Round 3
Subcontests
(4)
#2
1
Hide problems
recurrences
a sequence
(
a
n
)
(a_n)
(
a
n
)
n
n
n
≥
1
\geq 1
≥
1
is defined by the following equations;
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
2
a_2=2
a
2
=
2
,
a
3
=
1
a_3=1
a
3
=
1
,
a
2
n
−
1
a_{2n-1}
a
2
n
−
1
a
2
n
a_{2n}
a
2
n
=
a
2
a_2
a
2
a
2
n
−
3
a_{2n-3}
a
2
n
−
3
+
(
a
2
a
2
n
−
3
+
a
4
a
2
n
−
5
.
.
.
.
.
+
a
2
n
−
2
a
1
)
(a_2a_{2n-3}+a_4a_{2n-5}.....+a_{2n-2}a_1)
(
a
2
a
2
n
−
3
+
a
4
a
2
n
−
5
.....
+
a
2
n
−
2
a
1
)
for
n
n
n
≥
2
\geq 2
≥
2
n
a
2
n
na_{2n}
n
a
2
n
a
2
n
+
1
a_{2n+1}
a
2
n
+
1
=
a
2
a_2
a
2
a
2
n
−
2
a_{2n-2}
a
2
n
−
2
+
(
a
2
a
2
n
−
2
+
a
4
a
2
n
−
4
.
.
.
.
.
+
a
2
n
−
2
a
2
)
(a_2a_{2n-2}+a_4a_{2n-4}.....+a_{2n-2}a_2)
(
a
2
a
2
n
−
2
+
a
4
a
2
n
−
4
.....
+
a
2
n
−
2
a
2
)
for
n
n
n
≥
2
\geq 2
≥
2
find
a
2020
a_{2020}
a
2020
#4
1
Hide problems
twin primes and diophatines
let
p
p
p
and
q
=
p
+
2
q=p+2
q
=
p
+
2
be twin primes. consider the diophantine equation
(
+
)
(+)
(
+
)
given by
n
!
+
p
q
2
=
(
m
p
)
2
n!+pq^2=(mp)^2
n
!
+
p
q
2
=
(
m
p
)
2
m
≥
1
m\geq1
m
≥
1
,
n
≥
1
n\geq1
n
≥
1
i. if
m
=
p
m=p
m
=
p
,find the value of
p
p
p
. ii. how many solution quadruple
(
p
,
q
,
m
,
n
)
(p,q,m,n)
(
p
,
q
,
m
,
n
)
does
(
+
)
(+)
(
+
)
have ?
#3
1
Hide problems
Points on integer x-axis
given any 3 distinct points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
on the integer coordinates of the x-axis,the following operation is allowed:A point say
X
X
X
is reflected over another point say
Y
Y
Y
. Note that after each operation only one among three points is moved. we perform these operations till 2 out of the 3 points coincide. let
N
=
N
(
X
,
Y
,
Z
)
N=N(X,Y,Z)
N
=
N
(
X
,
Y
,
Z
)
denote the minimum number of operations before we are forced to stop.(this could happen in different ways). show that there are at most
2
N
2^N
2
N
coordinates that point
X
X
X
could end up if we are forced to stop after
N
N
N
operations
#1
1
Hide problems
parallel lines
in
A
B
C
ABC
A
BC
let
E
E
E
and
F
F
F
be points on line
A
C
AC
A
C
and
A
B
AB
A
B
respectively such that
B
E
BE
BE
is parallel to
C
F
CF
CF
. suppose that the circumcircle of
B
C
E
BCE
BCE
meet
A
B
AB
A
B
again at
F
′
F'
F
′
and the circumcircle of
B
C
F
BCF
BCF
meets
A
C
AC
A
C
again at
E
′
E'
E
′
. show that
B
E
′
BE'
B
E
′
Is parallel to
C
F
′
CF'
C
F
′
.