MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
1998 South africa National Olympiad
1998 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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n squares
You are given
n
n
n
squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?
5
1
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Gcd of binomial coefficients
Prove that
gcd
(
(
n
1
)
,
(
n
2
)
,
…
,
(
n
n
−
1
)
)
\gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)}
g
cd
(
(
1
n
)
,
(
2
n
)
,
…
,
(
n
−
1
n
)
)
is a prime if
n
n
n
is a power of a prime, and 1 otherwise.
4
1
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Friends
In a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.
3
1
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PQR similar to DGH
A
,
B
,
C
,
D
,
E
A,\ B,\ C,\ D,\ E
A
,
B
,
C
,
D
,
E
and
F
F
F
lie (in that order) on the circumference of a circle. The chords
A
D
,
B
E
AD,\ BE
A
D
,
BE
and
C
F
CF
CF
are concurrent.
P
,
Q
P,\ Q
P
,
Q
and
R
R
R
are the midpoints of
A
D
,
B
E
AD,\ BE
A
D
,
BE
and
C
F
CF
CF
respectively. Two further chords
A
G
∥
B
E
AG \parallel BE
A
G
∥
BE
and
A
H
∥
C
F
AH \parallel CF
A
H
∥
CF
are drawn. Show that
P
Q
R
PQR
PQR
is similar to
D
G
H
DGH
D
G
H
.
2
1
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Maximum value
Find the maximum value of
sin
2
α
+
sin
2
β
+
sin
2
γ
\sin{2\alpha} + \sin{2\beta} + \sin{2\gamma}
sin
2
α
+
sin
2
β
+
sin
2
γ
where
α
,
β
\alpha,\beta
α
,
β
and
γ
\gamma
γ
are positive and
α
+
β
+
γ
=
18
0
∘
\alpha + \beta + \gamma = 180^{\circ}
α
+
β
+
γ
=
18
0
∘
.
1
1
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Rational?
Is
log
10
8
\log_{10}{8}
lo
g
10
8
rational?