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Contests
International Contests
Pan-African Shortlist
2019 Pan-African Shortlist
2019 Pan-African Shortlist
Part of
Pan-African Shortlist
Subcontests
(6)
N5
1
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Each term in the sequence has a new prime divisor
Let
n
>
1
n > 1
n
>
1
be a positive integer. Prove that every term of the sequence
n
−
1
,
n
n
−
1
,
n
n
2
−
1
,
n
n
3
−
1
,
…
n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots
n
−
1
,
n
n
−
1
,
n
n
2
−
1
,
n
n
3
−
1
,
…
has a prime divisor that does not divide any of the previous terms.
G3
1
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Line through intersection of diagonals of cyclic quad and mipoint of side
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with its diagonals intersecting at
E
E
E
. Let
M
M
M
be the midpoint of
A
B
AB
A
B
. Suppose that
M
E
ME
ME
is perpendicular to
C
D
CD
C
D
. Show that either
A
C
AC
A
C
is perpendicular to
B
D
BD
B
D
, or
A
B
AB
A
B
is parallel to
C
D
CD
C
D
.
C2
1
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Replace two numbers with sum of digits of their sum
On the board, we write the integers
1
,
2
,
3
,
…
,
2019
1, 2, 3, \dots, 2019
1
,
2
,
3
,
…
,
2019
. At each minute, we pick two numbers on the board
a
a
a
and
b
b
b
, delete them, and write down the number
s
(
a
+
b
)
s(a + b)
s
(
a
+
b
)
instead, where
s
(
n
)
s(n)
s
(
n
)
denotes the sum of the digits of the integer
n
n
n
. Let
N
N
N
be the last number on the board at the end. [*] Is it possible to get
N
=
19
N = 19
N
=
19
? [*] Is it possible to get
N
=
15
N = 15
N
=
15
?
C1
1
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How many non-attacking pawns can be placed on a $n \times n$ chessboard?
A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an
n
×
n
n \times n
n
×
n
chessboard such that no two pawns attack each other?
A5
1
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Sequence appears to grow quite quickly. Is it always positive?
Let a sequence
(
a
i
)
i
=
10
∞
(a_i)_{i=10}^{\infty}
(
a
i
)
i
=
10
∞
be defined as follows: [*]
a
10
a_{10}
a
10
is some positive integer, which can of course be written in base 10. [*] For
i
≥
10
i \geq 10
i
≥
10
if
a
i
>
0
a_i > 0
a
i
>
0
, let
b
i
b_i
b
i
be the positive integer whose base-
(
i
+
1
)
(i + 1)
(
i
+
1
)
representation is the same as
a
i
a_i
a
i
's base-
i
i
i
representation. Then let
a
i
+
1
=
b
i
−
1
a_{i + 1} = b_i - 1
a
i
+
1
=
b
i
−
1
. If
a
i
=
0
a_i = 0
a
i
=
0
,
a
i
+
1
=
0
a_{i + 1} = 0
a
i
+
1
=
0
. For example, if
a
10
=
11
a_{10} = 11
a
10
=
11
, then
b
10
=
1
1
11
(
=
1
2
10
)
b_{10} = 11_{11} (= 12_{10})
b
10
=
1
1
11
(
=
1
2
10
)
;
a
11
=
1
1
11
−
1
=
1
0
11
(
=
1
1
10
)
a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})
a
11
=
1
1
11
−
1
=
1
0
11
(
=
1
1
10
)
;
b
11
=
1
0
12
(
=
1
2
10
)
b_{11} = 10_{12} (= 12_{10})
b
11
=
1
0
12
(
=
1
2
10
)
;
a
12
=
11
a_{12} = 11
a
12
=
11
.Does there exist
a
10
a_{10}
a
10
such that
a
i
a_i
a
i
is strictly positive for all
i
≥
10
i \geq 10
i
≥
10
?
A3
1
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Functional Equation: $f(x^2) - yf(y) = f(x + y)(f(x) - y)$.
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
2
)
−
y
f
(
y
)
=
f
(
x
+
y
)
(
f
(
x
)
−
y
)
f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y)
f
(
x
2
)
−
y
f
(
y
)
=
f
(
x
+
y
)
(
f
(
x
)
−
y
)
for all real numbers
x
x
x
and
y
y
y
.