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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2020 South Africa National Olympiad
2020 South Africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
6
1
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SAMO Problem 6: Repeatedly substract ceiling of square root until 0 is obtained
Marjorie is the drum major of the world's largest marching band, with more than one million members. She would like the band members to stand in a square formation. To this end, she determines the smallest integer
n
n
n
such that the band would fit in an
n
×
n
n \times n
n
×
n
square, and lets the members form rows of
n
n
n
people. However, she is dissatisfied with the result, since some empty positions remain. Therefore, she tells the entire first row of
n
n
n
members to go home and repeats the process with the remaining members. Her aim is to continue it until the band forms a perfect square, but as it happens, she does not succeed until the last members are sent home. Determine the smallest possible number of members in this marching band.
5
1
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SAMO Problem 5: Cyclic quadrilateral through points of two isosceles triangles
Let
A
B
C
ABC
A
BC
be a triangle, and let
T
T
T
be a point on the extension of
A
B
AB
A
B
beyond
B
B
B
, and
U
U
U
a point on the extension of
A
C
AC
A
C
beyond
C
C
C
, such that
B
T
=
C
U
BT = CU
BT
=
C
U
. Moreover, let
R
R
R
and
S
S
S
be points on the extensions of
A
B
AB
A
B
and
A
C
AC
A
C
beyond
A
A
A
such that
A
S
=
A
T
AS = AT
A
S
=
A
T
and
A
R
=
A
U
AR = AU
A
R
=
A
U
. Prove that
R
R
R
,
S
S
S
,
T
T
T
,
U
U
U
lie on a circle whose centre lies on the circumcircle of
A
B
C
ABC
A
BC
.
4
1
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SAMO Problem 4: Number of solutions to Diophantine equation
A positive integer
k
k
k
is said to be visionary if there are integers
a
>
0
a > 0
a
>
0
and
b
≥
0
b \geq 0
b
≥
0
such that
a
⋅
k
+
b
⋅
(
k
+
1
)
=
2020.
a \cdot k + b \cdot (k + 1) = 2020.
a
⋅
k
+
b
⋅
(
k
+
1
)
=
2020.
How many visionary integers are there?
3
1
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SAMO Problem 3: Solution to system of polynomial equations
If
x
x
x
,
y
y
y
,
z
z
z
are real numbers satisfying \begin{align*} (x + 1)(y + 1)(z + 1) & = 3 \\ (x + 2)(y + 2)(z + 2) & = -2 \\ (x + 3)(y + 3)(z + 3) & = -1, \end{align*} find the value of
(
x
+
20
)
(
y
+
20
)
(
z
+
20
)
.
(x + 20)(y + 20)(z + 20).
(
x
+
20
)
(
y
+
20
)
(
z
+
20
)
.
2
1
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SAMO Problem 2: Area of rhombus overlapping a square
Let
S
S
S
be a square with sides of length
2
2
2
and
R
R
R
be a rhombus with sides of length
2
2
2
and angles measuring
6
0
∘
60^\circ
6
0
∘
and
12
0
∘
120^\circ
12
0
∘
. These quadrilaterals are arranged to have the same centre and the diagonals of the rhombus are parallel to the sides of the square. Calculate the area of the region on which the figures overlap.
1
1
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SAMO Problem 1: Smallest positive integer with given number of divisors
Find the smallest positive multiple of
20
20
20
with exactly
20
20
20
positive divisors.