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Problems
Contests
International Contests
Pan African
2021 Pan-African
2021 Pan-African
Part of
Pan African
Subcontests
(6)
3
1
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PAMO Problem 3: A sequence of integers which is eventually pseudo geometric
Let
(
a
i
)
i
∈
N
(a_i)_{i\in \mathbb{N}}
(
a
i
)
i
∈
N
and
(
p
i
)
i
∈
N
(p_i)_{i\in \mathbb{N}}
(
p
i
)
i
∈
N
be two sequences of positive integers such that the following conditions hold:
∙
a
1
≥
2
\bullet ~~a_1\ge 2
∙
a
1
≥
2
.
∙
p
n
\bullet~~ p_n
∙
p
n
is the smallest prime divisor of
a
n
a_n
a
n
for every integer
n
≥
1
n\ge 1
n
≥
1
∙
a
n
+
1
=
a
n
+
a
n
p
n
\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}
∙
a
n
+
1
=
a
n
+
p
n
a
n
for every integer
n
≥
1
n\ge 1
n
≥
1
Prove that there is a positive integer
N
N
N
such that
a
n
+
3
=
3
a
n
a_{n+3}=3a_n
a
n
+
3
=
3
a
n
for every integer
n
>
N
n>N
n
>
N
2
1
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PAMO Problem 2: Equality of angles in a geometry problem with tangents
Let
Γ
\Gamma
Γ
be a circle,
P
P
P
be a point outside it, and
A
A
A
and
B
B
B
the intersection points between
Γ
\Gamma
Γ
and the tangents from
P
P
P
to
Γ
\Gamma
Γ
. Let
K
K
K
be a point on the line
A
B
AB
A
B
, distinct from
A
A
A
and
B
B
B
and let
T
T
T
be the second intersection point of
Γ
\Gamma
Γ
and the circumcircle of the triangle
P
B
K
PBK
PB
K
.Also, let
P
′
P'
P
′
be the reflection of
P
P
P
in point
A
A
A
. Show that
∠
P
B
T
=
∠
P
′
K
A
\angle PBT=\angle P'KA
∠
PBT
=
∠
P
′
K
A
1
1
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PAMO Problem 1: Number of trapezoids satisfying a certain condition
Let
n
n
n
be an integer greater than
3
3
3
. A square of side length
n
n
n
is divided by lines parallel to each side into
n
2
n^2
n
2
squares of length
1
1
1
. Find the number of convex trapezoids which have vertices among the vertices of the
n
2
n^2
n
2
squares of side length
1
1
1
, have side lengths less than or equal
3
3
3
and have area equal to
2
2
2
Note: Parallelograms are trapezoids.
6
1
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PAMO Problem 6: Two tangent circles
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid which is not a parallelogram, such that
A
D
AD
A
D
is parallel to
B
C
BC
BC
. Let
O
=
B
D
∩
A
C
O=BD\cap AC
O
=
B
D
∩
A
C
and
S
S
S
be the second intersection of the circumcircles of triangles
A
O
B
AOB
A
OB
and
D
O
C
DOC
D
OC
. Prove that the circumcircles of triangles
A
S
D
ASD
A
S
D
and
B
S
C
BSC
BSC
are tangent.
5
1
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PAMO Problem 5: Functional equation
Find all functions
f
f
f
:
:
:
R
→
R
\mathbb{R} \rightarrow \mathbb{R}
R
→
R
such that
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
:
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
=
f
(
x
2
)
+
f
(
y
2
)
+
2
f
(
x
y
)
(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
=
f
(
x
2
)
+
f
(
y
2
)
+
2
f
(
x
y
)
4
1
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PAMO Problem 4: Two integers satisfying two divisbilities
Find all integers
m
m
m
and
n
n
n
such that
m
2
+
n
n
2
−
m
\frac{m^2+n}{n^2-m}
n
2
−
m
m
2
+
n
and
n
2
+
m
m
2
−
n
\frac{n^2+m}{m^2-n}
m
2
−
n
n
2
+
m
are both integers.