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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2022 Nigerian Senior MO Round 2
2022 Nigerian Senior MO Round 2
Part of
Nigerian Senior Mathematics Olympiad Round 2
Subcontests
(6)
Problem 6
1
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Tricky diophantine equation
Let
k
,
l
,
m
,
n
k, l, m, n
k
,
l
,
m
,
n
be positive integers. Given that
k
+
l
+
m
+
n
=
k
m
=
l
n
k+l+m+n=km=ln
k
+
l
+
m
+
n
=
km
=
l
n
, find all possible values of
k
+
l
+
m
+
n
k+l+m+n
k
+
l
+
m
+
n
.
Problem 5
1
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How many paths?
For how many paths comsisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram below, is the word
OLYMPIADS
\textup{OLYMPIADS}
OLYMPIADS
spelled out as the path is traversed from beginning to end?\begin{tabular}{ccccccccccccccccc}& & & & & & & & O & & & & & & & &\\ & & & & & & & O & L & O & & & & & & &\\ & & & & & & O & L & Y & L & O & & & & & &\\ & & & & & O & L & Y & M & Y & L & O & & & & &\\ & & & & O & L & Y & M & P & M & Y & L & O & & & &\\ & & & O & L & Y & M & P & I & P & M & Y & L & O & & &\\ & & O & L & Y & M & P & I & A & I & P & M & Y & L & O & &\\ & O & L & Y & M & P & I & A & D & A & I & P & M & Y & L & O &\\ O & L & Y & M & P & I & A & D & S & D & A & I & P & M & Y & L & O \end{tabular}
Problem 4
1
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Two linked sequences
Define sequence
(
a
n
)
n
=
1
∞
(a_{n})_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
by
a
1
=
a
2
=
a
3
=
1
a_1=a_2=a_3=1
a
1
=
a
2
=
a
3
=
1
and
a
n
+
3
=
a
n
+
1
+
a
n
a_{n+3}=a_{n+1}+a_{n}
a
n
+
3
=
a
n
+
1
+
a
n
for all
n
≥
1
n \geq 1
n
≥
1
. Also, define sequence
(
b
n
)
n
=
1
∞
(b_{n})_{n=1}^{\infty}
(
b
n
)
n
=
1
∞
by
b
1
=
b
2
=
b
3
=
b
4
=
b
5
=
1
b_1=b_2=b_3=b_4=b_5=1
b
1
=
b
2
=
b
3
=
b
4
=
b
5
=
1
and
b
n
+
5
=
b
n
+
4
+
b
n
b_{n+5}=b_{n+4}+b_{n}
b
n
+
5
=
b
n
+
4
+
b
n
for all
n
≥
1
n \geq 1
n
≥
1
. Prove that
∃
N
∈
N
\exists N \in \mathbb{N}
∃
N
∈
N
such that
a
n
=
b
n
+
1
+
b
n
−
8
a_n = b_{n+1} + b_{n-8}
a
n
=
b
n
+
1
+
b
n
−
8
for all
n
≥
N
n \geq N
n
≥
N
.
Problem 3
1
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Real lengths?
In triangle
A
B
C
ABC
A
BC
,
A
D
AD
A
D
and
A
E
AE
A
E
trisect
∠
B
A
C
\angle BAC
∠
B
A
C
. The lengths of
B
D
,
D
E
BD, DE
B
D
,
D
E
and
E
C
EC
EC
are
1
,
3
1, 3
1
,
3
and
5
5
5
respectively. Find the length of
A
C
AC
A
C
.
Problem 2
1
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Show 4 points are concyclic
Let
G
G
G
be the centroid of
△
A
B
C
\triangle ABC
△
A
BC
and let
D
,
E
D, E
D
,
E
and
F
F
F
be the midpoints of the line segments
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
respectively. Suppose the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
meets
A
D
AD
A
D
again at
X
X
X
, the circumcircle of
△
D
E
F
\triangle DEF
△
D
EF
meets
B
E
BE
BE
again at
Y
Y
Y
and the circumcircle of
△
D
E
F
\triangle DEF
△
D
EF
meets
C
F
CF
CF
again at
Z
Z
Z
. Show that
G
,
X
,
Y
G, X, Y
G
,
X
,
Y
and
Z
Z
Z
are concyclic.
Problem 1
1
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Diophantine equation
Find all integer solutions of the equation
x
y
+
5
x
−
3
y
=
27
xy+5x-3y=27
x
y
+
5
x
−
3
y
=
27
.