MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2019 South Africa National Olympiad
2019 South Africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
Hide problems
SAMO Problem 6: Diophantine equation $20^m - 10m^2 + 1 = 19^n$
Determine all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of non-negative integers that satisfy the equation
2
0
m
−
10
m
2
+
1
=
1
9
n
.
20^m - 10m^2 + 1 = 19^n.
2
0
m
−
10
m
2
+
1
=
1
9
n
.
5
1
Hide problems
SAMO Problem 5: Functional equation based on expansion of $(a + b + c)^3$
Find all functions
f
:
Z
→
Z
f : \mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that
f
(
a
3
)
+
f
(
b
3
)
+
f
(
c
3
)
+
3
f
(
a
+
b
)
f
(
b
+
c
)
f
(
c
+
a
)
=
(
f
(
a
+
b
+
c
)
)
3
f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3
f
(
a
3
)
+
f
(
b
3
)
+
f
(
c
3
)
+
3
f
(
a
+
b
)
f
(
b
+
c
)
f
(
c
+
a
)
=
(
f
(
a
+
b
+
c
))
3
for all integers
a
,
b
,
c
a, b, c
a
,
b
,
c
.
4
1
Hide problems
SAMO Problem 4: Rectangles on chess board with all black corners
The squares of an
8
×
8
8 \times 8
8
×
8
board are coloured alternatingly black and white. A rectangle consisting of some of the squares of the board is called important if its sides are parallel to the sides of the board and all its corner squares are coloured black. The side lengths can be anything from
1
1
1
to
8
8
8
squares. On each of the
64
64
64
squares of the board, we write the number of important rectangles in which it is contained. The sum of the numbers on the black squares is
B
B
B
, and the sum of the numbers on the white squares is
W
W
W
. Determine the difference
B
−
W
B - W
B
−
W
.
3
1
Hide problems
SAMO Problem 3: Product of lengths in triangle with $45^\circ$ angle
Let
A
A
A
,
B
B
B
,
C
C
C
be points on a circle whose centre is
O
O
O
and whose radius is
1
1
1
, such that
∠
B
A
C
=
4
5
∘
\angle BAC = 45^\circ
∠
B
A
C
=
4
5
∘
. Lines
A
C
AC
A
C
and
B
O
BO
BO
(possibly extended) intersect at
D
D
D
, and lines
A
B
AB
A
B
and
C
O
CO
CO
(possibly extended) intersect at
E
E
E
. Prove that
B
D
⋅
C
E
=
2
BD \cdot CE = 2
B
D
⋅
CE
=
2
.
2
1
Hide problems
SAMO Problem 2: Sums of consecutive natural numbers
We have a deck of
90
90
90
cards that are numbered from
10
10
10
to
99
99
99
(all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?
1
1
Hide problems
SAMO Problem 1: Divisibility with exponents
Determine all positive integers
a
a
a
for which
a
a
a^a
a
a
is divisible by
2
0
19
20^{19}
2
0
19
.