MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2000 South africa National Olympiad
2000 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Tiling of a 4 x n rectangle
Let
A
n
A_n
A
n
be the number of ways to tile a
4
×
n
4 \times n
4
×
n
rectangle using
2
×
1
2 \times 1
2
×
1
tiles. Prove that
A
n
A_n
A
n
is divisible by 2 if and only if
A
n
A_n
A
n
is divisible by 3.
5
1
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Functional equation with integers
Find all functions
f
:
Z
→
Z
f: \mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
(where
Z
\mathbb{Z}
Z
is the set of all integers) such that
2000
f
(
f
(
x
)
)
−
3999
f
(
x
)
+
1999
x
=
0
for all
x
∈
Z
.
2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}.
2000
f
(
f
(
x
))
−
3999
f
(
x
)
+
1999
x
=
0
for all
x
∈
Z
.
4
1
Hide problems
Square and perimeter of triangle
A
B
C
D
ABCD
A
BC
D
is a square of side 1.
P
P
P
and
Q
Q
Q
are points on
A
B
AB
A
B
and
B
C
BC
BC
such that
P
D
Q
^
=
4
5
∘
\widehat{PDQ} = 45^{\circ}
P
D
Q
=
4
5
∘
. Find the perimeter of
Δ
P
B
Q
\Delta PBQ
Δ
PBQ
.
3
1
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Sequence of integers
Let
c
≥
1
c \geq 1
c
≥
1
be an integer, and define the sequence
a
1
,
a
2
,
a
3
,
…
a_1,\ a_2,\ a_3,\ \dots
a
1
,
a
2
,
a
3
,
…
by
a
1
=
2
,
a
n
+
1
=
c
a
n
+
(
c
2
−
1
)
(
a
n
2
−
4
)
for
n
=
1
,
2
,
3
,
…
.
\begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned}
a
1
a
n
+
1
=
2
,
=
c
a
n
+
(
c
2
−
1
)
(
a
n
2
−
4
)
for
n
=
1
,
2
,
3
,
…
.
Prove that
a
n
a_n
a
n
is an integer for all
n
n
n
.
2
1
Hide problems
Quartic equation
Solve for
x
x
x
, given
36
x
4
+
36
x
3
−
7
x
2
−
6
x
+
1
=
0
36x^4 + 36x^3 - 7x^2 - 6x + 1 = 0
36
x
4
+
36
x
3
−
7
x
2
−
6
x
+
1
=
0
.
1
1
Hide problems
Prime?
A number
x
n
x_n
x
n
of the form 10101...1 has
n
n
n
ones. Find all
n
n
n
such that
x
n
x_n
x
n
is prime.