MathDB
Problems
Contests
International Contests
Pan African
2019 Pan-African
2019 Pan-African
Part of
Pan African
Subcontests
(6)
6
1
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PAMO Problem 6: Binomial coefficient not divisible by 5
Find the
2019
2019
2019
th strictly positive integer
n
n
n
such that
(
2
n
n
)
\binom{2n}{n}
(
n
2
n
)
is not divisible by
5
5
5
.
5
1
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PAMO Problem 5: Broken line covering centres of squares
A square is divided into
N
2
N^2
N
2
equal smaller non-overlapping squares, where
N
≥
3
N \geq 3
N
≥
3
. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself).[*] Show that it is possible to find a broken line composed of
4
4
4
segments for
N
=
3
N = 3
N
=
3
. [*] Find the minimum number of segments in this broken line for arbitrary
N
N
N
.
4
1
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PAMO Problem 4: Perpendicular lines
The tangents to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
B
B
B
and
C
C
C
meet at
D
D
D
. The circumcircle of
△
B
C
D
\triangle BCD
△
BC
D
meets sides
A
C
AC
A
C
and
A
B
AB
A
B
again at
E
E
E
and
F
F
F
respectively. Let
O
O
O
be the circumcentre of
△
A
B
C
\triangle ABC
△
A
BC
. Show that
A
O
AO
A
O
is perpendicular to
E
F
EF
EF
.
3
1
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PAMO Problem 3: Circumcentres coincide implies equilateral
Let
A
B
C
ABC
A
BC
be a triangle, and
D
D
D
,
E
E
E
,
F
F
F
points on the segments
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
respectively such that
B
D
D
C
=
C
E
E
A
=
A
F
F
B
.
\frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}.
D
C
B
D
=
E
A
CE
=
FB
A
F
.
Show that if the centres of the circumscribed circles of the triangles
D
E
F
DEF
D
EF
and
A
B
C
ABC
A
BC
coincide, then
A
B
C
ABC
A
BC
is an equilateral triangle.
2
1
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PAMO Problem 2: Product and sum of primes
Let
k
k
k
be a positive integer. Consider
k
k
k
not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of
k
k
k
?
1
1
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PAMO Problem 1: Linear recurrence relation
Let
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
be a sequence of real numbers defined as follows:[*]
a
0
=
3
a_0 = 3
a
0
=
3
,
a
1
=
2
a_1 = 2
a
1
=
2
, and
a
2
=
12
a_2 = 12
a
2
=
12
; and [*]
2
a
n
+
3
−
a
n
+
2
−
8
a
n
+
1
+
4
a
n
=
0
2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0
2
a
n
+
3
−
a
n
+
2
−
8
a
n
+
1
+
4
a
n
=
0
for
n
≥
0
n \geq 0
n
≥
0
.Show that
a
n
a_n
a
n
is always a strictly positive integer.