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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2008 South africa National Olympiad
2008 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Find all pairs of functions
Find all function pairs
(
f
,
g
)
(f,g)
(
f
,
g
)
where each
f
f
f
and
g
g
g
is a function defined on the integers and with values, such that, for all integers
a
a
a
and
b
b
b
,
f
(
a
+
b
)
=
f
(
a
)
g
(
b
)
+
g
(
a
)
f
(
b
)
g
(
a
+
b
)
=
g
(
a
)
g
(
b
)
−
f
(
a
)
f
(
b
)
.
f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).
f
(
a
+
b
)
=
f
(
a
)
g
(
b
)
+
g
(
a
)
f
(
b
)
g
(
a
+
b
)
=
g
(
a
)
g
(
b
)
−
f
(
a
)
f
(
b
)
.
5
1
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Perpendiculars from the orthocentre to angle bisectors
Triangle
A
B
C
ABC
A
BC
has orthocentre
H
H
H
. The feet of the perpendiculars from
H
H
H
to the internal and external bisectors of
A
^
\hat{A}
A
^
are
P
P
P
and
Q
Q
Q
respectively. Prove that
P
P
P
is on the line that passes through
Q
Q
Q
and the midpoint of
B
C
BC
BC
. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)
4
1
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Which card was at the top of the pile?
A pack of
2008
2008
2008
cards, numbered from
1
1
1
to
2008
2008
2008
, is shuffled in order to play a game in which each move has two steps:(i) the top card is placed at the bottom;(ii) the new top card is removed.It turns out that the cards are removed in the order
1
,
2
,
…
,
2008
1,2,\dots,2008
1
,
2
,
…
,
2008
. Which card was at the top before the game started?
3
1
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Product (a+b) > 8 Product (a+b-c)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
and determine when equality occurs.
2
1
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Products of sides and diagonals in a quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with the property that
A
B
AB
A
B
extended and
C
D
CD
C
D
extended intersect at a right angle. Prove that
A
C
⋅
B
D
>
A
D
⋅
B
C
AC\cdot BD>AD\cdot BC
A
C
⋅
B
D
>
A
D
⋅
BC
.
1
1
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How many small divisors does this power have?
Determine the number of positive divisors of
200
8
8
2008^8
200
8
8
that are less than
200
8
4
2008^4
200
8
4
.