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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
1999 South africa National Olympiad
1999 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Lattice points - Number of paths
You are at a point
(
a
,
b
)
(a,b)
(
a
,
b
)
and you need to reach another point
(
c
,
d
)
(c,d)
(
c
,
d
)
. Both points are below the line
x
=
y
x = y
x
=
y
and have integer coordinates. You can move in steps of length 1, either upwards of to the right, but you may not move to a point on the line
x
=
y
x = y
x
=
y
. How many different paths are there?
5
1
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Denominators are powers of 3
Let
S
S
S
be the set of all rational numbers whose denominators are powers of 3. Let
a
a
a
,
b
b
b
and
c
c
c
be given non-zero real numbers. Determine all real-valued functions
f
f
f
that are defined for
x
∈
S
x \in S
x
∈
S
, satisfy
f
(
x
)
=
a
f
(
3
x
)
+
b
f
(
3
x
−
1
)
+
c
f
(
3
x
−
2
)
if
0
≤
x
≤
1
,
f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1,
f
(
x
)
=
a
f
(
3
x
)
+
b
f
(
3
x
−
1
)
+
c
f
(
3
x
−
2
)
if
0
≤
x
≤
1
,
and are zero elsewhere.
4
1
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L(p) - 1 is divisible by p if p is prime
The sequence
L
1
,
L
2
,
L
3
,
…
L_1,\ L_2,\ L_3,\ \dots
L
1
,
L
2
,
L
3
,
…
is defined by
L
1
=
1
,
L
2
=
3
,
L
n
=
L
n
−
1
+
L
n
−
2
for
n
>
2.
L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2.
L
1
=
1
,
L
2
=
3
,
L
n
=
L
n
−
1
+
L
n
−
2
for
n
>
2.
Prove that
L
p
−
1
L_p - 1
L
p
−
1
is divisible by
p
p
p
if
p
p
p
is prime.
3
1
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Centre of (CKL) lies on (BCD)
The bisector of
∠
B
A
D
\angle{BAD}
∠
B
A
D
in the parallellogram
A
B
C
D
ABCD
A
BC
D
intersects the lines
B
C
BC
BC
and
C
D
CD
C
D
at the points
K
K
K
and
L
L
L
respectively. Prove that the centre of the circle passing through the points
C
,
K
C,\ K
C
,
K
and
L
L
L
lies on the circle passing through the points
B
,
C
B,\ C
B
,
C
and
D
D
D
.
2
1
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Construct the square...
A
,
B
,
C
A,\ B,\ C
A
,
B
,
C
and
D
D
D
are points on a given straight line, in that order. Show how to construct a square
P
Q
R
S
PQRS
PQRS
, with all of
P
,
Q
,
R
P,\ Q,\ R
P
,
Q
,
R
and
S
S
S
on the same side of
A
D
AD
A
D
, such that
A
,
B
,
C
A,\ B,\ C
A
,
B
,
C
and
D
D
D
lie on
P
Q
,
S
R
,
Q
R
PQ,\ SR,\ QR
PQ
,
SR
,
QR
and
P
S
PS
PS
produced respectively.
1
1
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Triangles with perimeter 1999
How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?