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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2006 South africa National Olympiad
2006 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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This average of floor functions fluctuates
Consider the function
f
f
f
defined by
f
(
n
)
=
1
n
(
⌊
n
1
⌋
+
⌊
n
2
⌋
+
⋯
+
⌊
n
n
⌋
)
f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )
f
(
n
)
=
n
1
(
⌊
1
n
⌋
+
⌊
2
n
⌋
+
⋯
+
⌊
n
n
⌋
)
for all positive integers
n
n
n
. (Here
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
.) Prove that(a)
f
(
n
+
1
)
>
f
(
n
)
f(n+1)>f(n)
f
(
n
+
1
)
>
f
(
n
)
for infinitely many
n
n
n
.(b)
f
(
n
+
1
)
<
f
(
n
)
f(n+1)<f(n)
f
(
n
+
1
)
<
f
(
n
)
for infinitely many
n
n
n
.
5
1
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Subsets without consecutive elements
Find the number of subsets
X
X
X
of
{
1
,
2
,
…
,
10
}
\{1,2,\dots,10\}
{
1
,
2
,
…
,
10
}
such that
X
X
X
contains at least two elements and such that no two elements of
X
X
X
differ by
1
1
1
.
4
1
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Obtuse isosceles triangle
In triangle
A
B
C
ABC
A
BC
,
A
B
=
A
C
AB=AC
A
B
=
A
C
and
B
A
^
C
=
10
0
∘
B\hat{A}C=100^\circ
B
A
^
C
=
10
0
∘
. Let
D
D
D
be on
A
C
AC
A
C
such that
A
B
^
D
=
C
B
^
D
A\hat{B}D=C\hat{B}D
A
B
^
D
=
C
B
^
D
. Prove that
A
D
+
D
B
=
B
C
AD+DB=BC
A
D
+
D
B
=
BC
.
3
1
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Squares that end on 196
Determine all positive integers whose squares end in
196
196
196
.
2
1
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What is the largest that this angle can be?
Triangle
A
B
C
ABC
A
BC
has
B
C
=
1
BC=1
BC
=
1
and
A
C
=
2
AC=2
A
C
=
2
. What is the maximum possible value of
A
^
\hat{A}
A
^
.
1
1
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Simplify the fraction
Reduce the fraction
2121212121210
1121212121211
\frac{2121212121210}{1121212121211}
1121212121211
2121212121210
to its simplest form.