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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 3
2019 Nigerian Senior MO Round 3
2019 Nigerian Senior MO Round 3
Part of
Nigerian Senior Mathematics Olympiad Round 3
Subcontests
(4)
4
1
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find the least possible area of the initial rectangular grid before the cuttings
A rectangular grid whose side lengths are integers greater than
1
1
1
is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than
1
1
1
are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings.Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here
3
1
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5^{2019} / \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}
Show that
5
2019
∣
Σ
k
=
1
5
2019
3
g
c
d
(
5
2019
,
k
)
5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}
5
2019
∣
Σ
k
=
1
5
2019
3
g
c
d
(
5
2019
,
k
)
2
1
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Triangle inequality
Let
a
b
c
abc
ab
c
be real numbers satisfying
a
b
+
b
c
+
c
a
=
1
ab+bc+ca=1
ab
+
b
c
+
c
a
=
1
. Show that
∣
a
−
b
∣
∣
1
+
c
2
∣
\frac{|a-b|}{|1+c^2|}
∣1
+
c
2
∣
∣
a
−
b
∣
+
∣
b
−
c
∣
∣
1
+
a
2
∣
\frac{|b-c|}{|1+a^2|}
∣1
+
a
2
∣
∣
b
−
c
∣
>
=
>=
>=
∣
c
−
a
∣
∣
1
+
b
2
∣
\frac{|c-a|}{|1+b^2|}
∣1
+
b
2
∣
∣
c
−
a
∣
1
1
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Cyclic quadrilaterals
Let the altitude from
A
A
A
and
B
B
B
of triangle
A
B
C
ABC
A
BC
meet the circumcircle of
A
B
C
ABC
A
BC
again at
D
D
D
and
E
E
E
respectively. Let
D
E
DE
D
E
meet
A
C
AC
A
C
and
B
C
BC
BC
at
P
P
P
and
Q
Q
Q
respectively. Show that
A
B
Q
P
ABQP
A
BQP
is cyclic