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Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2016 Lusophon Mathematical Olympiad
2016 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
5
1
Hide problems
lusophone sequence: 1, 1 (x prime or +1), .., .. (x prime or +1), 2016
A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number
1
1
1
. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number (
2
,
3
,
5
,
7
,
11
,
.
.
.
2,3,5,7,11, ...
2
,
3
,
5
,
7
,
11
,
...
) or add
1
1
1
. (iii) The last term of the sequence is the number
2016
2016
2016
. For example:
1
→
×
11
11
→
×
61
671
→
+
1
672
→
×
3
2016
1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016
1
→
×
11
11
→
×
61
671
→
+
1
672
→
×
3
2016
How many Lusophone sequences exist in which (as in the example above) the add
1
1
1
operation was used exactly once and not multiplied twice by the same prime number?
3
1
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4th largest root of integer P(x), so it divides sum of coefficients =3, gingado
Suppose a real number
a
a
a
is a root of a polynomial with integer coefficients
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
. Let
G
=
∣
a
n
∣
+
∣
a
n
−
1
∣
+
.
.
.
+
∣
a
1
∣
+
∣
a
0
∣
G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|
G
=
∣
a
n
∣
+
∣
a
n
−
1
∣
+
...
+
∣
a
1
∣
+
∣
a
0
∣
. We say that
G
G
G
is a gingado of
a
a
a
. For example, as
2
2
2
is root of
P
(
x
)
=
x
2
−
x
−
2
P(x)=x^2-x-2
P
(
x
)
=
x
2
−
x
−
2
,
G
=
∣
1
∣
+
∣
−
1
∣
+
∣
−
2
∣
=
4
G=|1|+|-1|+|-2|=4
G
=
∣1∣
+
∣
−
1∣
+
∣
−
2∣
=
4
, we say that
4
4
4
is a gingado of
2
2
2
. What is the fourth largest real number
a
a
a
such that
3
3
3
is a gingado of
a
a
a
?
1
1
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smallest no of distinct prime factors in product of 10 distinct coprime integers
Consider
10
10
10
distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of
10
10
10
numbers?
4
1
Hide problems
8 football teams, all once, first 4 end with 15 pts,other 4 w N pts,12 draws
8
8
8
CPLP football teams competed in a championship in which each team played one and only time with each of the other teams. In football, each win is worth
3
3
3
points, each draw is worth
1
1
1
point and the defeated team does not score. In that championship four teams were in first place with
15
15
15
points and the others four came in second with
N
N
N
points each. Knowing that there were
12
12
12
draws throughout the championship, determine
N
N
N
.
6
1
Hide problems
Power of 2
Source: Lusophon MO 2016 Prove that any positive power of
2
2
2
can be written as:
5
x
y
−
x
2
−
2
y
2
5xy-x^2-2y^2
5
x
y
−
x
2
−
2
y
2
where
x
x
x
and
y
y
y
are odd numbers.
2
1
Hide problems
Center and midpoint
The circle
ω
1
\omega_1
ω
1
intersects the circle
ω
2
\omega_2
ω
2
in the points
A
A
A
and
B
B
B
, a tangent line to this circles intersects
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
in the points
E
E
E
and
F
F
F
respectively. Suppose that
A
A
A
is inside of the triangle
B
E
F
BEF
BEF
, let
H
H
H
be the orthocenter of
B
E
F
BEF
BEF
and
M
M
M
is the midpoint of
B
H
BH
B
H
. Prove that the centers of the circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
and the point
M
M
M
are collinears.