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National and Regional Contests
South Africa Contests
South Africa National Olympiad
2013 South africa National Olympiad
2013 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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Geometric Inequality - South African Maths Olympiad
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
C
≠
B
C
AC \neq BC
A
C
=
BC
, and let
O
O
O
be the circumcentre and
F
F
F
the foot of the altitude through
C
C
C
. Furthermore, let
X
X
X
and
Y
Y
Y
be the feet of the perpendiculars dropped from
A
A
A
and
B
B
B
respectively to (the extension of)
C
O
CO
CO
. The line
F
O
FO
FO
intersects the circumcircle of
F
X
Y
FXY
FX
Y
a second time at
P
P
P
. Prove that
O
P
<
O
F
OP<OF
OP
<
OF
.
5
1
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Neighbouring coins on a chess board
Some coins are placed on a
20
×
13
20 \times 13
20
×
13
board. Two coins are called neighbours if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?
4
1
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Functional equation with polynomials
Determine all pairs of polynomials
f
f
f
and
g
g
g
with real coefficients such that
x
2
⋅
g
(
x
)
=
f
(
g
(
x
)
)
.
x^2 \cdot g(x) = f(g(x)).
x
2
⋅
g
(
x
)
=
f
(
g
(
x
))
.
3
1
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SAMO 2013 Senior Round 3 Problem 3 - Tangent to Circumcircle
Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by
l
l
l
, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of
l
l
l
and t. Show that CE and t are perpendicular to each other.
2
1
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SAMO 2013 Senior Round 3 Problem 2
A is a two-digit number and B is a three-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).
1
1
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SAMO 2013 Senior Round 3 Problem 1
2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 (and before 10000)?