MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2005 South africa National Olympiad
2005 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
Hide problems
Explicit formula for this sequence
Consider the increasing sequence
1
,
2
,
4
,
5
,
7
,
9
,
10
,
12
,
14
,
16
,
17
,
19
,
…
1,2,4,5,7,9,10,12,14,16,17,19,\dots
1
,
2
,
4
,
5
,
7
,
9
,
10
,
12
,
14
,
16
,
17
,
19
,
…
of positive integers, obtained by concatenating alternating blocks
{
1
}
,
{
2
,
4
}
,
{
5
,
7
,
9
}
,
{
10
,
12
,
14
,
16
}
,
…
\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots
{
1
}
,
{
2
,
4
}
,
{
5
,
7
,
9
}
,
{
10
,
12
,
14
,
16
}
,
…
of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the
n
n
n
-th element of the sequence is given by
2
n
−
⌊
1
+
8
n
−
7
2
⌋
.
2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.
2
n
−
⌊
2
1
+
8
n
−
7
⌋
.
(Here
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
.)
5
1
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There exists a k satisfying this inequality
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
be positive numbers with product equal to 1. Prove that there exists a
k
∈
{
1
,
2
,
…
,
n
}
k\in\{1,2,\dots,n\}
k
∈
{
1
,
2
,
…
,
n
}
such that
x
k
k
+
x
1
+
x
2
+
⋯
+
x
k
≥
1
−
1
2
n
.
\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.
k
+
x
1
+
x
2
+
⋯
+
x
k
x
k
≥
1
−
n
2
1
.
4
1
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Intersection of a cevian with the incircle
The inscribed circle of triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at
D
D
D
,
E
E
E
and
F
F
F
respectively. Let
Q
Q
Q
denote the other point of intersection of
A
D
AD
A
D
and the inscribed circle. Prove that
E
Q
EQ
EQ
extended passes through the midpoint of
A
F
AF
A
F
if and only if
A
C
=
B
C
AC = BC
A
C
=
BC
.
3
1
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Useable pairs of boots
A warehouse contains
175
175
175
boots of size
8
8
8
,
175
175
175
boots of size
9
9
9
and
200
200
200
boots of size
10
10
10
. Of these
550
550
550
boots,
250
250
250
are for the left foot and
300
300
300
for the right foot. Let
n
n
n
denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.)(a) Is
n
=
50
n=50
n
=
50
possible?(b) Is
n
=
51
n=51
n
=
51
possible?
2
1
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The largest fraction less than 16/23
Let
F
F
F
be the set of all fractions
m
/
n
m/n
m
/
n
where
m
m
m
and
n
n
n
are positive integers with
m
+
n
≤
2005
m+n\le 2005
m
+
n
≤
2005
. Find the largest number
a
a
a
in
F
F
F
such that
a
<
16
/
23
a < 16/23
a
<
16/23
.
1
1
Hide problems
Prove that the sum is always the same
Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same.
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
\begin{array}{|c|c|c|c|c|}\hline 1&4&7&10&13\\ \hline 16&19&22&25&28\\ \hline 31&34&37&40&43\\ \hline 46&49&52&55&58\\ \hline 61&64&67&70&73\\ \hline \end{array}
1
16
31
46
61
4
19
34
49
64
7
22
37
52
67
10
25
40
55
70
13
28
43
58
73