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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2012 South africa National Olympiad
2012 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Functional inequality
Find all functions
f
:
N
→
R
f:\mathbb{N}\to\mathbb{R}
f
:
N
→
R
such that
f
(
k
m
)
+
f
(
k
n
)
−
f
(
k
)
f
(
m
n
)
≥
1
f(km)+f(kn)-f(k)f(mn)\ge 1
f
(
km
)
+
f
(
kn
)
−
f
(
k
)
f
(
mn
)
≥
1
for all
k
,
m
,
n
∈
N
k,m,n\in\mathbb{N}
k
,
m
,
n
∈
N
.
5
1
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Triangles have the same centroid
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
. We denote its orthocentre by
H
H
H
, its circumcentre by
O
O
O
and the midpoint of
B
C
BC
BC
by
D
D
D
. The extensions of
H
D
HD
HD
and
A
O
AO
A
O
meet in
P
P
P
. Prove that triangles
A
H
P
AHP
A
H
P
and
A
B
C
ABC
A
BC
have the same centroid.
4
1
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x^k + px = y^k
Let
p
p
p
and
k
k
k
be positive integers such that
p
p
p
is prime and
k
>
1
k>1
k
>
1
. Prove that there is at most one pair
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers such that
x
k
+
p
x
=
y
k
x^k+px=y^k
x
k
+
p
x
=
y
k
.
3
1
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Colored points on a circle
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number
1
1
1
, an arc between a red and a blue point is assigned a number
2
2
2
, and an arc between a blue and a green point is assigned a number
3
3
3
. The arcs between two points of the same colour are assigned a number
0
0
0
. What is the greatest possible sum of all the numbers assigned to the arcs?
2
1
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SAMO 2012 Senior Round 3 Problem 2
Let
A
B
C
D
ABCD
A
BC
D
be a square and
X
X
X
a point such that
A
A
A
and
X
X
X
are on opposite sides of
C
D
CD
C
D
. The lines
A
X
AX
A
X
and
B
X
BX
BX
intersect
C
D
CD
C
D
in
Y
Y
Y
and
Z
Z
Z
respectively. If the area of
A
B
C
D
ABCD
A
BC
D
is
1
1
1
and the area of
X
Y
Z
XYZ
X
Y
Z
is
2
3
\frac{2}{3}
3
2
, determine the length of
Y
Z
YZ
Y
Z
1
1
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SAMO 2012 Senior Round 3 Problem 1
Given that
1
+
3
+
5
+
⋯
+
(
2
n
−
1
)
2
+
4
+
6
+
⋯
+
(
2
n
)
=
2011
2012
\frac{1+3+5+\cdots+(2n-1)}{2+4+6+\cdots+(2n)}=\frac{2011}{2012}
2
+
4
+
6
+
⋯
+
(
2
n
)
1
+
3
+
5
+
⋯
+
(
2
n
−
1
)
=
2012
2011
, determine n.