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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 3
2022 Nigerian MO round 3
2022 Nigerian MO round 3
Part of
Nigerian Senior Mathematics Olympiad Round 3
Subcontests
(4)
Problem 3
1
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Tiling problem (Combinatorics or Number Theory?)
A unit square is removed from the corner of an
n
×
n
n \times n
n
×
n
grid, where
n
≥
2
n \geq 2
n
≥
2
. Prove that the remainder can be covered by copies of the figures of
3
3
3
or
5
5
5
unit squares depicted in the drawing below. [asy] import geometry;draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0));draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy]Note: Every square must be covered once and figures must not go over the bounds of the grid.
Problem 1
1
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Permutation sequence
Integer sequence
(
x
n
)
(x_{n})
(
x
n
)
is defined as follows;
x
1
=
1
x_{1} = 1
x
1
=
1
, and for each integer
n
≥
1
n \geq 1
n
≥
1
,
x
n
+
1
x_{n+1}
x
n
+
1
is equal to the largest number that can be obtained by permutation of the digits of
x
n
+
2
x_{n}+2
x
n
+
2
. Find the smallest
n
n
n
for which the decimal representation of
x
n
x_{n}
x
n
contains exactly
2022
2022
2022
digits
Problem 2
1
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Real functions
If
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
satisfies
f
(
x
2
+
f
(
y
)
)
=
y
+
x
f
(
x
)
f(x^2 +f(y))=y+xf(x)
f
(
x
2
+
f
(
y
))
=
y
+
x
f
(
x
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
, find
f
(
x
)
f(x)
f
(
x
)
.
Problem 4
1
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Two tangents and bisection
Let
P
T
PT
PT
and
P
B
PB
PB
be two tangents to a circle,
T
T
T
and
B
B
B
on the circle.
A
B
AB
A
B
is the diameter of the circle through
B
B
B
and
T
H
TH
T
H
is the perpendicular from
T
T
T
to
A
B
AB
A
B
,
H
H
H
on
A
B
AB
A
B
. Prove that
A
P
AP
A
P
bisects
T
H
TH
T
H
.