MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2001 South africa National Olympiad
2001 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
Hide problems
System with unique solution
The unknown real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
satisfy
x
1
<
x
2
<
⋯
<
x
n
,
x_1 < x_2 < \cdots < x_n,
x
1
<
x
2
<
⋯
<
x
n
,
where
n
≥
3
n \geq 3
n
≥
3
. The numbers
s
s
s
,
t
t
t
and
d
1
,
d
2
,
…
,
d
n
−
2
d_1,d_2,\dots,d_{n - 2}
d
1
,
d
2
,
…
,
d
n
−
2
are given, such that
s
=
∑
i
=
1
n
x
i
,
t
=
∑
i
=
1
n
x
i
2
,
d
i
=
x
i
+
2
−
x
i
,
i
=
1
,
2
,
…
,
n
−
2.
\begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned}
s
t
d
i
=
i
=
1
∑
n
x
i
,
=
i
=
1
∑
n
x
i
2
,
=
x
i
+
2
−
x
i
,
i
=
1
,
2
,
…
,
n
−
2.
For which
n
n
n
is this information always sufficient to determine
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
uniquely?
5
1
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Sequence of quadrilaterals!
Starting from a given cyclic quadrilateral
Q
0
\mathcal{Q}_0
Q
0
, a sequence of quadrilaterals is constructed so that
Q
k
+
1
\mathcal{Q}_{k + 1}
Q
k
+
1
is the circumscribed quadrilateral of
Q
k
\mathcal{Q}_k
Q
k
for
k
=
0
,
1
,
…
k = 0,1,\dots
k
=
0
,
1
,
…
. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral
A
B
C
D
ABCD
A
BC
D
has sides that are tangent to the circumcircle of
A
B
C
D
ABCD
A
BC
D
at
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
.) Prove that the sequence always terminates, except when
Q
0
\mathcal{Q}_0
Q
0
is a square.
4
1
Hide problems
2n blue and red points
n
n
n
red and
n
n
n
blue points on a plane are given so that no three of the
2
n
2n
2
n
points are collinear. Prove that it is always possible to split up the points into
n
n
n
pairs, with one red and one blue point in each pair, so that no two of the
n
n
n
line segments which connect the two members of a pair intersect.
3
1
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x^1919, x^1960 and x^2001
For a certain real number
x
x
x
, the differences between
x
1919
x^{1919}
x
1919
,
x
1960
x^{1960}
x
1960
and
x
2001
x^{2001}
x
2001
are all integers. Prove that
x
x
x
is an integer.
2
1
Hide problems
Complicated equations :)
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers that satisfy
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
y
z
=
4
(
x
+
y
+
z
)
.
\begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned}
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
yz
=
4
(
x
+
y
+
z
)
.
1
1
Hide problems
p/2 < AC + BD < p
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral with perimeter
p
p
p
. Prove that
1
2
p
<
A
C
+
B
D
<
p
.
\dfrac{1}{2}p < AC + BD < p.
2
1
p
<
A
C
+
B
D
<
p
.
(A polygon is convex if all of its interior angles are less than
18
0
∘
180^\circ
18
0
∘
.)