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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 3
2021 Nigerian MO Round 3
2021 Nigerian MO Round 3
Part of
Nigerian Senior Mathematics Olympiad Round 3
Subcontests
(6)
Problem 6
1
Hide problems
Deleting products
Let
m
≤
n
m \leq n
m
≤
n
be natural numbers. Starting with the product
t
=
m
⋅
(
m
+
1
)
⋅
(
m
+
2
)
⋅
⋯
⋅
n
t=m\cdot (m+1) \cdot (m+2) \cdot \cdots \cdot n
t
=
m
⋅
(
m
+
1
)
⋅
(
m
+
2
)
⋅
⋯
⋅
n
, let
T
m
,
n
T_{m, n}
T
m
,
n
be the sum of products that can be obtained from deleting from
t
t
t
pairs of consecutive integers (this includes
t
t
t
itself). In the case where all the numbers are deleted, we assume the number
1
1
1
.For example,
T
2
,
7
=
2
⋅
3
⋅
4
⋅
5
⋅
6
⋅
7
+
2
⋅
3
⋅
4
⋅
5
+
2
⋅
3
⋅
4
⋅
7
+
2
⋅
3
⋅
6
⋅
7
+
2
⋅
5
⋅
6
⋅
7
+
4
⋅
5
⋅
6
⋅
7
+
2
⋅
3
+
2
⋅
5
+
2
⋅
7
+
4
⋅
7
+
6
⋅
7
+
1
=
5040
+
120
+
168
+
252
+
420
+
840
+
6
+
10
+
14
+
20
+
28
+
42
+
1
=
6961
T_{2, 7} = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 \cdot 4 \cdot 5 + 2 \cdot 3 \cdot 4 \cdot 7 + 2 \cdot 3 \cdot 6 \cdot 7 + 2 \cdot 5 \cdot 6 \cdot 7 + 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 + 2 \cdot 5 + 2 \cdot 7 + 4 \cdot 7 + 6 \cdot 7 + 1 = 5040 + 120 + 168 + 252 + 420 + 840 + 6 + 10 + 14 + 20 + 28 + 42 + 1 = 6961
T
2
,
7
=
2
⋅
3
⋅
4
⋅
5
⋅
6
⋅
7
+
2
⋅
3
⋅
4
⋅
5
+
2
⋅
3
⋅
4
⋅
7
+
2
⋅
3
⋅
6
⋅
7
+
2
⋅
5
⋅
6
⋅
7
+
4
⋅
5
⋅
6
⋅
7
+
2
⋅
3
+
2
⋅
5
+
2
⋅
7
+
4
⋅
7
+
6
⋅
7
+
1
=
5040
+
120
+
168
+
252
+
420
+
840
+
6
+
10
+
14
+
20
+
28
+
42
+
1
=
6961
.Taking
T
n
+
1
,
n
=
1
T_{n+1, n} = 1
T
n
+
1
,
n
=
1
.Show that
T
m
,
n
+
1
=
T
m
,
k
−
1
⋅
T
k
+
2
,
n
+
1
+
T
m
,
k
⋅
T
k
+
1
,
n
+
1
T_{m, n+1}=T_{m, k-1} \cdot T_{k+2, n+1} + T_{m, k} \cdot T_{k+1, n+1}
T
m
,
n
+
1
=
T
m
,
k
−
1
⋅
T
k
+
2
,
n
+
1
+
T
m
,
k
⋅
T
k
+
1
,
n
+
1
for all
1
≤
m
≤
k
≤
n
1 \leq m \leq k \leq n
1
≤
m
≤
k
≤
n
.
Problem 5
1
Hide problems
Derivatives, polynomials and functions
Let
f
(
x
)
=
P
(
x
)
Q
(
x
)
f(x)=\frac{P(x)}{Q(x)}
f
(
x
)
=
Q
(
x
)
P
(
x
)
, where
P
(
x
)
,
Q
(
x
)
P(x), Q(x)
P
(
x
)
,
Q
(
x
)
are two non-constant polynomials with no common zeros and
P
(
0
)
=
P
(
1
)
=
0
P(0)=P(1)=0
P
(
0
)
=
P
(
1
)
=
0
. Suppose
f
(
x
)
f
(
1
x
)
=
f
(
x
)
+
f
(
1
x
)
f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)
f
(
x
)
f
(
x
1
)
=
f
(
x
)
+
f
(
x
1
)
for infinitely many values of
x
x
x
. a) Show that
deg
(
P
)
<
deg
(
Q
)
\text{deg}(P)<\text{deg}(Q)
deg
(
P
)
<
deg
(
Q
)
. b) Show that
P
′
(
1
)
=
2
Q
′
(
1
)
−
deg
(
Q
)
⋅
Q
(
1
)
P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)
P
′
(
1
)
=
2
Q
′
(
1
)
−
deg
(
Q
)
⋅
Q
(
1
)
.Here,
P
′
(
x
)
P'(x)
P
′
(
x
)
denotes the derivative of
P
(
x
)
P(x)
P
(
x
)
as usual.
Problem 4
1
Hide problems
Multiplication magic square
In the multiplication magic square below,
l
,
m
,
n
,
p
,
q
,
r
,
s
,
t
,
u
l, m, n, p, q, r, s, t, u
l
,
m
,
n
,
p
,
q
,
r
,
s
,
t
,
u
are positive integers. The product of any three numbers in any row, column or diagonal is equal to a constant
k
k
k
, where
k
k
k
is a number between
11
,
000
11, 000
11
,
000
and
12
,
500
12, 500
12
,
500
. Find the value of
k
k
k
. \begin{tabular}{|l|l|l|} \hline
l
l
l
&
m
m
m
&
n
n
n
\\ \hline
p
p
p
&
q
q
q
&
r
r
r
\\ \hline
s
s
s
&
t
t
t
&
u
u
u
\\ \hline \end{tabular}
Problem 3
1
Hide problems
Fifth and sitxh power of prime
Find all pairs of natural numbers
(
p
,
n
)
(p, n)
(
p
,
n
)
with
p
p
p
prime such that
p
6
+
p
5
+
n
3
+
n
=
n
5
+
n
2
p^6+p^5+n^3+n=n^5+n^2
p
6
+
p
5
+
n
3
+
n
=
n
5
+
n
2
.
Problem 2
1
Hide problems
Cyclic quads and collinear points
Let
B
,
C
,
D
,
E
B, C, D, E
B
,
C
,
D
,
E
be four pairwise distinct collinear points and let
A
A
A
be a point not on ine
B
C
BC
BC
. Now, let the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
meet
A
D
AD
A
D
and
A
E
AE
A
E
respectively again at
F
F
F
and
G
G
G
. Show that
D
E
F
G
DEFG
D
EFG
is cyclic if and only if
A
B
=
A
C
AB=AC
A
B
=
A
C
.
Problem 1
1
Hide problems
Triples of primes
Find all triples of primes
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
such that
p
q
=
2021
+
r
3
p^q=2021+r^3
p
q
=
2021
+
r
3
.