MathDB
Problems
Contests
National and Regional Contests
Nigeria Contests
2024 Nigerian MO Round 3
2024 Nigerian MO Round 3
Part of
Nigeria Contests
Subcontests
(4)
Problem 1
1
Hide problems
long modular arithmetic
Find the value of
(
2
40
+
1
2
41
+
2
3
42
+
6
7
43
+
8
7
44
)
45
!
+
46
m
o
d
11
(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11
(
2
40
+
1
2
41
+
2
3
42
+
6
7
43
+
8
7
44
)
45
!
+
46
mod
11
(variation but same answer) 3
Problem 4
1
Hide problems
polygon combinatorics
In an island shaped like a regular polygon of
n
n
n
sides, there are airports at each vertex of the island. The island would like to add
k
k
k
new airports into the interior of the island, but it must follow the following rules:\\
1
1
1
. It must be in the interior of the island (none on borders).\\
2
2
2
. No two airports can be at the exact same location.\\
3
3
3
. Every triple of
1
1
1
new and
2
2
2
old airports must form an isoceles triangle.\\
4
4
4
. No three airports can be collinear.\\Find the maximum value of
k
k
k
for each
n
n
n
[hide=Harder Version]Replace
1
1
1
new and
2
2
2
old with
1
1
1
old and
2
2
2
new.
Problem 3
1
Hide problems
cos(a+b)/cos(a-b)
Let
A
B
C
ABC
A
BC
be a triangle, and let
O
O
O
be its circumcenter. Let
C
O
‾
∩
A
B
≡
D
\overline{CO}\cap AB\equiv D
CO
∩
A
B
≡
D
. Let
∠
B
A
C
=
α
\angle BAC=\alpha
∠
B
A
C
=
α
, and
∠
C
B
A
=
β
\angle CBA=\beta
∠
CB
A
=
β
. Prove that
O
D
O
C
=
∣
cos
(
α
+
β
)
cos
(
α
−
β
)
∣
\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|
OC
O
D
=
cos
(
α
−
β
)
cos
(
α
+
β
)
\\For clarification,
C
O
‾
\overline{CO}
CO
represents the line
C
O
CO
CO
, and
A
C
AC
A
C
represents the segment
A
C
AC
A
C
. Cases in which
D
D
D
doesn't exist should be ignored.
Problem 2
1
Hide problems
x^3+y^2|x^2+y^3
Prove that there exist infinitely many distinct positive integers,
x
x
x
and
y
y
y
, such that
x
3
+
y
2
∣
x
2
+
y
3
x^3+y^2|x^2+y^3
x
3
+
y
2
∣
x
2
+
y
3