MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2023 South Africa National Olympiad
2023 South Africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(6)
5
1
Hide problems
closing down k South Adrican airports
South Adrican Magical Flights (SAMF) operates flights between South Adrican airports. If there is a flight from airport
A
A
A
to airpost
B
B
B
, there will be also a flight from
B
B
B
to
A
A
A
. The SAMF headquarters are located in Kimberley. Every airport that is served by Kimberley can be reached from Kimberley in precisely one way. This way of reaching Kimberley may involve stopping at other airports on the way. (For example, it may happen that you can get to Kimberley by flying from Durban to Bloemfontein and then from to Bloemfontein to Kimberley. In that case there is no other way to get from Durban to Kimberley. For example, there would be no direct Hight from Durban to Kimberley.) An airport (other than Kimberley) is called terminal if there are flights to (and from) precisely one other airport. Suppose that there are
t
t
t
terminal airports. Due to budget cuts, SAMF decides to close down
k
k
k
of the airports. It should still be possible to reach each of the remaining airports from Kimberley. Let
C
C
C
be the number of choices for the
k
k
k
destinations that are discontinued. Prove that
t
!
k
!
(
t
−
k
)
≤
C
≤
(
t
+
k
−
1
)
!
k
!
(
t
−
1
)
!
.
\frac{t!}{k!(t-k)} \le C \le \frac{(t+k-1)!}{k!(t-1)!} .
k
!
(
t
−
k
)
t
!
≤
C
≤
k
!
(
t
−
1
)!
(
t
+
k
−
1
)!
.
6
1
Hide problems
concurrency related to erected rectangles
Let
A
B
I
H
ABIH
A
B
I
H
,
B
D
E
C
BDEC
B
D
EC
and
A
C
F
G
ACFG
A
CFG
be arbitrary rectangles constructed (externally) on the sides of triangle
A
B
C
ABC
A
BC
.Choose point
S
S
S
outside rectangle
A
B
I
H
ABIH
A
B
I
H
(on the opposite side as triangle
A
B
C
ABC
A
BC
) such that
∠
S
H
I
=
∠
F
A
C
\angle SHI=\angle FAC
∠
S
H
I
=
∠
F
A
C
and
∠
H
I
S
=
∠
E
B
C
\angle HIS=\angle EBC
∠
H
I
S
=
∠
EBC
.Prove that the lines
F
I
,
E
H
FI,EH
F
I
,
E
H
and
C
S
CS
CS
are concurrent(i.e., the three lines intersect in one point).
4
1
Hide problems
Integers expressible as product of elements of an interesting set
Let
A
A
A
be a set of real numbers satisfying the following: (a)
(
n
2
+
1
)
∈
A
\sqrt(n^2+1) \in A
(
n
2
+
1
)
∈
A
for all positive integers
n
n
n
, (b) if
x
∈
A
x \in A
x
∈
A
and
y
∈
A
y \in A
y
∈
A
, then
x
−
y
∈
A
x-y \in A
x
−
y
∈
A
. Prove that every integer can be written as a product of two different elements in
A
A
A
.
3
1
Hide problems
GCD and LCM
Consider
2
2
2
positive integers
a
,
b
a,b
a
,
b
such that
a
+
2
b
=
2020
a+2b=2020
a
+
2
b
=
2020
. (a) Determine the largest possible value of the greatest common divisor of
a
a
a
and
b
b
b
. (b) Determine the smallest possible value of the least common multiple of
a
a
a
and
b
b
b
.
2
1
Hide problems
Cyclic quad
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral with
∠
B
A
D
=
9
0
∘
\angle BAD=90^\circ
∠
B
A
D
=
9
0
∘
and
∠
A
B
C
>
9
0
∘
\angle ABC>90^\circ
∠
A
BC
>
9
0
∘
.
A
B
AB
A
B
is extended to a point
E
E
E
such that
∠
A
E
C
=
9
0
∘
\angle AEC=90^\circ
∠
A
EC
=
9
0
∘
.If
A
B
=
7
,
B
E
=
9
,
AB=7,BE=9,
A
B
=
7
,
BE
=
9
,
and
E
C
=
12
EC=12
EC
=
12
,calculate
A
D
AD
A
D
.
1
1
Hide problems
Equations
If
x
x
x
and
y
y
y
are positive real numbers such that
(
2
x
)
+
(
y
)
=
13
\sqrt(2x)+\sqrt(y)=13
(
2
x
)
+
(
y
)
=
13
and
(
8
x
)
+
(
9
y
)
=
35
\sqrt(8x)+\sqrt(9y)=35
(
8
x
)
+
(
9
y
)
=
35
, calculate
20
x
+
23
y
20x+23y
20
x
+
23
y
.