MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2014 South africa National Olympiad
2014 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
Hide problems
Proportional areas mean that points lie on a line
Let
O
O
O
be the centre of a two-dimensional coordinate system, and let
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots ,A_n
A
1
,
A
2
,
…
,
A
n
be points in the first quadrant and
B
1
,
B
2
,
…
,
B
m
B_1, B_2, \ldots , B_m
B
1
,
B
2
,
…
,
B
m
points in the second quadrant. We associate numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots , a_n
a
1
,
a
2
,
…
,
a
n
to the points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots ,A_n
A
1
,
A
2
,
…
,
A
n
and numbers
b
1
,
b
2
,
…
,
b
m
b_1, b_2, \ldots, b_m
b
1
,
b
2
,
…
,
b
m
to the points
B
1
,
B
2
,
…
,
B
m
B_1, B_2, \ldots , B_m
B
1
,
B
2
,
…
,
B
m
, respectively. It turns out that the area of triangle
O
A
j
B
k
OA_jB_k
O
A
j
B
k
is always equal to the product
a
j
b
k
a_jb_k
a
j
b
k
, for any
j
j
j
and
k
k
k
. Show that either all the
A
j
A_j
A
j
or all the
B
k
B_k
B
k
lie on a single line through
O
O
O
.
5
1
Hide problems
A green, blue and white colouring of a board.
Let
n
>
1
n > 1
n
>
1
be an integer. An
n
×
n
n \times n
n
×
n
-square is divided into
n
2
n^2
n
2
unit squares. Of these unit squares,
n
n
n
are coloured green and
n
n
n
are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row?
4
1
Hide problems
The function ax+gcd(a,x)+lcm(a,x)
(a) Let
a
,
x
,
y
a,x,y
a
,
x
,
y
be positive integers. Prove: if
x
≠
y
x\ne y
x
=
y
, the also
a
x
+
gcd
(
a
,
x
)
+
lcm
(
a
,
x
)
≠
a
y
+
gcd
(
a
,
y
)
+
lcm
(
a
,
y
)
.
ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).
a
x
+
g
cd
(
a
,
x
)
+
lcm
(
a
,
x
)
=
a
y
+
g
cd
(
a
,
y
)
+
lcm
(
a
,
y
)
.
(b) Show that there are no two positive integers
a
a
a
and
b
b
b
such that
a
b
+
gcd
(
a
,
b
)
+
lcm
(
a
,
b
)
=
2014.
ab+\gcd(a,b)+\text{lcm}(a,b)=2014.
ab
+
g
cd
(
a
,
b
)
+
lcm
(
a
,
b
)
=
2014.
3
1
Hide problems
Angles of obtuse triangle with given lines parallel
In obtuse triangle
A
B
C
ABC
A
BC
, with the obtuse angle at
A
A
A
, let
D
D
D
,
E
E
E
,
F
F
F
be the feet of the altitudes through
A
A
A
,
B
B
B
,
C
C
C
respectively.
D
E
DE
D
E
is parallel to
C
F
CF
CF
, and
D
F
DF
D
F
is parallel to the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. Find the angles of the triangle.
2
1
Hide problems
Given $\frac{a-b}{c-d}$ and $\frac{a-c}{b-d}=3$...
Given that \frac{a-b}{c-d}=2 \text{and} \frac{a-c}{b-d}=3 for certain real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, determine the value of
a
−
d
b
−
c
.
\frac{a-d}{b-c}.
b
−
c
a
−
d
.
1
1
Hide problems
Last 2 digits of a product of odd integers
Determine the last two digits of the product of the squares of all positive odd integers less than
2014
2014
2014
.