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Problems
Contests
National and Regional Contests
Mozambique Contests
2024 Mozambican National MO Selection Test
2024 Mozambican National MO Selection Test
Part of
Mozambique Contests
Subcontests
(3)
P2
1
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Circles and tangents
On a sheet divided into squares, each square measuring
2
c
m
2cm
2
c
m
, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.
P3
1
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Nice diophantine equation.
Find all triples of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that:
a
2
b
c
−
2
a
b
2
c
−
2
a
b
c
2
+
b
3
c
+
b
c
3
+
2
b
2
c
2
=
11
a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11
a
2
b
c
−
2
a
b
2
c
−
2
ab
c
2
+
b
3
c
+
b
c
3
+
2
b
2
c
2
=
11
P1
1
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Well payed security guard??
A school security guard works from Monday to Saturday from
7
:
30
a
m
7:30 am
7
:
30
am
to
12
:
00
p
m
12:00 pm
12
:
00
p
m
(
7
:
30
7:30
7
:
30
to
12
:
00
12:00
12
:
00
). He also works the night shift, from Monday to Friday from
6
p
m
6pm
6
p
m
to
10
p
m
10pm
10
p
m
(
18
:
00
18:00
18
:
00
to
22
:
00
22:00
22
:
00
) . He receives
75
M
T
75MT
75
MT
per hour, up to
40
40
40
hours of work per week. For the remaining hours of weekly work, he receives
95
M
T
95MT
95
MT
per hour. So, considering that a month has four weeks, what will be this security guard's monthly salary?