MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2010 South africa National Olympiad
2010 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
Hide problems
The largest minimum sum of entries in a table
Write either
1
1
1
or
−
1
-1
−
1
in each of the cells of a
(
2
n
)
×
(
2
n
)
(2n) \times (2n)
(
2
n
)
×
(
2
n
)
-table, in such a way that there are exactly
2
n
2
2n^2
2
n
2
entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be
M
M
M
. Determine the largest possible value of
M
M
M
.
5
1
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Lines intersecting at some angle alpha
(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle
α
\alpha
α
. Determine the smallest number of lines for which this is possible.(b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.
4
1
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Sum of (x_k)/(sqrt k)
Given
n
n
n
positive real numbers satisfying
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
x_1 \ge x_2 \ge \cdots \ge x_n \ge 0
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
and
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
1
x_1^2+x_2^2+\cdots+x_n^2=1
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
1
, prove that
x
1
1
+
x
2
2
+
⋯
+
x
n
n
≥
1.
\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.
1
x
1
+
2
x
2
+
⋯
+
n
x
n
≥
1.
3
1
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The product of an even number of consecutive integers
Determine all positive integers
n
n
n
such that
5
n
−
1
5^n - 1
5
n
−
1
can be written as a product of an even number of consecutive integers.
2
1
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Sum of squares of lengths in a triangle
Consider a triangle
A
B
C
ABC
A
BC
with
B
C
=
3
BC = 3
BC
=
3
. Choose a point
D
D
D
on
B
C
BC
BC
such that
B
D
=
2
BD = 2
B
D
=
2
. Find the value of
A
B
2
+
2
A
C
2
−
3
A
D
2
.
AB^2 + 2AC^2 - 3AD^2.
A
B
2
+
2
A
C
2
−
3
A
D
2
.
1
1
Hide problems
Sum of its digits plus the square of its units digit
For a positive integer
n
n
n
,
S
(
n
)
S(n)
S
(
n
)
denotes the sum of its digits and
U
(
n
)
U(n)
U
(
n
)
its unit digit. Determine all positive integers
n
n
n
with the property that
n
=
S
(
n
)
+
U
(
n
)
2
.
n = S(n) + U(n)^2.
n
=
S
(
n
)
+
U
(
n
)
2
.