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National and Regional Contests
South Africa Contests
South Africa National Olympiad
2007 South africa National Olympiad
2007 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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The South African Mathematical Olympiad 2007
Prove that it is not possible to write numbers
1
,
2
,
3
,
.
.
.
,
25
1,2,3,...,25
1
,
2
,
3
,
...
,
25
on the squares of
5
5
5
x
5
5
5
chessboard such that any neighboring numbers differ by at most
4
4
4
.
4
1
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The South African Mathematical Olympiad 2007
Let
A
B
C
ABC
A
BC
be a triangle and
P
Q
R
S
PQRS
PQRS
a square with
P
P
P
on
A
B
AB
A
B
,
Q
Q
Q
on
A
C
AC
A
C
, and
R
R
R
and
S
S
S
on
B
C
BC
BC
. Let
H
H
H
on
B
C
BC
BC
such that
A
H
AH
A
H
is the altitude of the triangle from
A
A
A
to base
B
C
BC
BC
. Prove that: (a) \frac{1}{AH} \plus{}\frac{1}{BC}\equal{}\frac{1}{PQ} (b) the area of
A
B
C
ABC
A
BC
is twice the area of
P
Q
R
S
PQRS
PQRS
iff AH\equal{}BC
3
1
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The South African Mathematical Olympiad 2007
In acute-angled triangle
A
B
C
ABC
A
BC
, the points
D
,
E
,
F
D,E,F
D
,
E
,
F
are on sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively such that \angle AFE \equal{} \angle BFD, \angle FDB \equal{} \angle EDC, \angle DEC \equal{} \angle FEA. Prove that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
.
5
1
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The South African Mathematical Olympiad 2007
Let
Z
Z
Z
and
R
R
R
denote the sets of integers and real numbers, respectively. Let
f
:
Z
→
R
f: Z \rightarrow R
f
:
Z
→
R
be a function satisfying: (i)
f
(
n
)
≥
0
f(n) \ge 0
f
(
n
)
≥
0
for all
n
∈
Z
n \in Z
n
∈
Z
(ii) f(mn)\equal{}f(m)f(n) for all
m
,
n
∈
Z
m,n \in Z
m
,
n
∈
Z
(iii) f(m\plus{}n) \le max(f(m),f(n)) for all
m
,
n
∈
Z
m,n \in Z
m
,
n
∈
Z
(a) Prove that
f
(
n
)
≤
1
f(n) \le 1
f
(
n
)
≤
1
for all
n
∈
Z
n \in Z
n
∈
Z
(b) Find a function
f
:
Z
→
R
f: Z \rightarrow R
f
:
Z
→
R
satisfying (i), (ii),(iii) and
0
<
f
(
2
)
<
1
0<f(2)<1
0
<
f
(
2
)
<
1
and f(2007) \equal{} 1
2
1
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The South African Mathematical Olympiad 2007
Consider the equation x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers. Determine the largest value of
b
b
b
for which this equation has exactly three distinct solutions, all of which are integers.
1
1
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The South African Mathematical Olympiad 2007
Determine whether \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}} is less than or greater than
3
10
\frac{3}{10}
10
3
.