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Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2021 Nigerian Senior MO Round 2
2021 Nigerian Senior MO Round 2
Part of
Nigerian Senior Mathematics Olympiad Round 2
Subcontests
(5)
3
1
Hide problems
board with lowest poaitive fractions
On a certain board, fractions are always written in their lowest form. Pionaj starts with 2 random positive fractions. After every minute,he replaces one of the previous 2 fractions (at random) with a new fraction that is equal to the sum of their numerators divided by the sum of their denominators. Given that he continues this indefinitely, show that eventually all the resulting fractions would be in their lowest forms even before writing them on the board(recall that he has to reduce each fration to their lowest form beore writing it on the board for the next operation). (for example starting with
15
7
\frac{15}{7}
7
15
and
10
3
\frac{10}{3}
3
10
he may replace it with
5
2
\frac{5}{2}
2
5
5
1
Hide problems
equal lengths ,tangents and angle bisectors
let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
E
E
E
,an interior point such that
A
B
=
A
D
=
A
E
=
B
C
AB=AD=AE=BC
A
B
=
A
D
=
A
E
=
BC
. Let
D
E
DE
D
E
meet the circumcircle of
B
E
C
BEC
BEC
again at
F
F
F
. Suppose a common tangent to the circumcircle of
B
E
C
BEC
BEC
and
D
E
C
DEC
D
EC
touch the circles at
F
F
F
and
G
G
G
respectively. Show that
G
E
GE
GE
is the external angle bisector of angle
B
E
F
BEF
BEF
4
1
Hide problems
maximum value is a rational number
let
x
1
x_1
x
1
,
x
2
x_2
x
2
....
x
6
x_6
x
6
be non-negative reals such that
x
1
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
=
1
x_1+x_2+x_3+x_4+x_5+x_6=1
x
1
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
=
1
and
x
1
x
3
x
5
x_1x_3x_5
x
1
x
3
x
5
+
x
2
x
4
x
6
x_2x_4x_6
x
2
x
4
x
6
≥
\geq
≥
1
540
\frac{1}{540}
540
1
. Let
p
p
p
and
q
q
q
be relatively prime integers such that
p
q
\frac{p}{q}
q
p
is the maximum value of
x
1
x
2
x
3
+
x
2
x
3
x
4
+
x
3
x
4
x
5
+
x
4
x
5
x
6
+
x
5
x
6
x
1
+
x
6
x
1
x
2
x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2
x
1
x
2
x
3
+
x
2
x
3
x
4
+
x
3
x
4
x
5
+
x
4
x
5
x
6
+
x
5
x
6
x
1
+
x
6
x
1
x
2
. Find
p
+
q
p+q
p
+
q
2
1
Hide problems
boxes and balls in a circle
N
N
N
boxes are arranged in a circle and are numbered
1
,
2
,
3
,
.
.
.
.
.
N
1,2,3,.....N
1
,
2
,
3
,
.....
N
In a clockwise direction. A ball is assigned a number from
1
,
2
,
3
,
.
.
.
.
N
{1,2,3,....N}
1
,
2
,
3
,
....
N
and is placed in one of the boxes.A round consist of the following; if the current number on the ball is
n
n
n
, the ball is moved
n
n
n
boxes in the clockwise direction and the number on the ball is changed to
n
+
1
n+1
n
+
1
if
n
<
N
n<N
n
<
N
and to
1
1
1
if
n
=
N
n=N
n
=
N
. Is it possible to choose
N
N
N
, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly
2020
2020
2020
rounds
1
1
Hide problems
Plotemy and regular nonagons
If
x
x
x
,
y
y
y
and
z
z
z
are the lengths of a side, a shortest diagonal and a longest diagonal respectively, of a regular nonagon. Write a correct equation consisting of the three lengths