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Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2021 South Africa National Olympiad
2021 South Africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Two player game involving summing squares
Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers
1
,
4
,
9
,
…
,
202
1
2
1, 4, 9, \dots, 2021^2
1
,
4
,
9
,
…
,
202
1
2
, and there is a whiteboard in front of them with the number
0
0
0
on it. Jacob chooses a number
x
2
x^2
x
2
from his list, removes it from his list, and replaces the number
W
W
W
on the whiteboard with
W
+
x
2
W + x^2
W
+
x
2
. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by
4
4
4
after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number
K
K
K
such that Jacob can guarantee to get at least
K
K
K
sheep by the end of the game, no matter how Laban plays?
5
1
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System of polynomial equations
Determine all polynomials
a
(
x
)
a(x)
a
(
x
)
,
b
(
x
)
b(x)
b
(
x
)
,
c
(
x
)
c(x)
c
(
x
)
,
d
(
x
)
d(x)
d
(
x
)
with real coefficients satisfying the simultaneous equations \begin{align*} b(x) c(x) + a(x) d(x) & = 0 \\ a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1. \end{align*}
4
1
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Altitude of triangle tangent to circle
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
≠
9
0
∘
\angle ABC \neq 90^\circ
∠
A
BC
=
9
0
∘
and
A
B
AB
A
B
its shortest side. Denote by
H
H
H
the intersection of the altitudes of triangle
A
B
C
ABC
A
BC
. Let
K
K
K
be the circle through
A
A
A
with centre
B
B
B
. Let
D
D
D
be the other intersection of
K
K
K
and
A
C
AC
A
C
. Let
K
K
K
intersect the circumcircle of
B
C
D
BCD
BC
D
again at
E
E
E
. If
F
F
F
is the intersection of
D
E
DE
D
E
and
B
H
BH
B
H
, show that
B
D
BD
B
D
is tangent to the circle through
D
D
D
,
F
F
F
, and
H
H
H
.
3
1
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Squares of primes sum to a power of 2
Determine the smallest integer
k
>
1
k > 1
k
>
1
such that there exist
k
k
k
distinct primes whose squares sum to a power of
2
2
2
.
2
1
Hide problems
Ratio of lengths in right-angled triangle
Let
P
A
B
PAB
P
A
B
and
P
B
C
PBC
PBC
be two similar right-angled triangles (in the same plane) with
∠
P
A
B
=
∠
P
B
C
=
9
0
∘
\angle PAB = \angle PBC = 90^\circ
∠
P
A
B
=
∠
PBC
=
9
0
∘
such that
A
A
A
and
C
C
C
lie on opposite sides of the line
P
B
PB
PB
. If
P
C
=
A
C
PC = AC
PC
=
A
C
, calculate the ratio
P
A
A
B
\frac{PA}{AB}
A
B
P
A
.
1
1
Hide problems
Numbers of a given form divisible by 11
Find the smallest and largest integers with decimal representation of the form
a
b
a
b
a
ababa
ababa
(
a
≠
0
a \neq 0
a
=
0
) that are divisible by
11
11
11
.