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Problems
Contests
National and Regional Contests
Brazil Contests
Mathematicians for Fun Olympiad
2020 OMpD
2020 OMpD
Part of
Mathematicians for Fun Olympiad
Subcontests
(4)
2
2
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3 bodybuilders fighting against a n-headed monster
Metadieu, Tercieu and Quartieu are three bodybuilder warriors who fight against an
n
n
n
-headed monster. Each of them can attack the monster according to the following rules:(1) Metadieu's attack consists of cutting off half of the monster's heads, then cutting off one more head. If the monster's number of heads is odd, Metadieu cannot attack;(2) Tercieu's attack consists of cutting off a third of the monster's heads, then cutting off two more heads. If the monster's number of heads is not a multiple of 3, Tercieu cannot attack;(3) Quartieu's attack consists of cutting off a quarter of the monster's heads, then cutting off three more heads. If the monster's number of heads is not a multiple of 4, Quartieu cannot attack;If none of the three warriors can attack the monster at some point, then it will devour our three heroes. The objective of the three warriors is to defeat the monster, and for that they need to cut off all its heads, one warrior attacking at a time.For what positive integer values of
n
n
n
is it possible for the three warriors to combine a sequence of attacks in order to defeat the monster?
Game with powers of 4
A pile of
2020
2020
2020
stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove
1
,
4
,
16
,
64
,
.
.
.
1, 4, 16, 64, ...
1
,
4
,
16
,
64
,
...
(any power of
4
4
4
) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?
1
2
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Diophantine with polynomial and exponential
Determine all pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that:
x
4
−
6
x
2
+
1
=
7
⋅
2
y
x^4 - 6x^2 + 1 = 7\cdot 2^y
x
4
−
6
x
2
+
1
=
7
⋅
2
y
find all possible values of a division
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers such that
a
+
b
+
c
=
0
a + b + c = 0
a
+
b
+
c
=
0
. Given that
a
3
+
b
3
+
c
3
≠
0
a^3 + b^3 + c^3 \neq 0
a
3
+
b
3
+
c
3
=
0
,
a
2
+
b
2
+
c
2
≠
0
a^2 + b^2 + c^2 \neq 0
a
2
+
b
2
+
c
2
=
0
, determine all possible values for:
a
5
+
b
5
+
c
5
(
a
3
+
b
3
+
c
3
)
(
a
2
+
b
2
+
c
2
)
\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}
(
a
3
+
b
3
+
c
3
)
(
a
2
+
b
2
+
c
2
)
a
5
+
b
5
+
c
5
3
2
Hide problems
incenter of tangential quad collinear with midpoints of diagonals
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral and let
Γ
\Gamma
Γ
be a circle of center
O
O
O
that is internally tangent to its four sides. If
M
M
M
is the midpoint of
A
C
AC
A
C
and
N
N
N
is the midpoint of
B
D
BD
B
D
, prove that
M
,
O
,
N
M,O, N
M
,
O
,
N
are collinear.
Polynomials are prime numbers
Determine all integers
n
n
n
such that both of the numbers:
∣
n
3
−
4
n
2
+
3
n
−
35
∣
and
∣
n
2
+
4
n
+
8
∣
|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|
∣
n
3
−
4
n
2
+
3
n
−
35∣
and
∣
n
2
+
4
n
+
8∣
are both prime numbers.
4
2
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Cute functional system
Let
R
+
\mathbb{R}^+
R
+
the set of positive real numbers. Determine all the functions
f
,
g
:
R
+
→
R
+
f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+
f
,
g
:
R
+
→
R
+
such that, for all positive real numbers
x
,
y
x, y
x
,
y
we have that
f
(
x
+
g
(
y
)
)
=
f
(
x
+
y
)
+
g
(
y
)
and
g
(
x
+
f
(
y
)
)
=
g
(
x
+
y
)
+
f
(
y
)
f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)
f
(
x
+
g
(
y
))
=
f
(
x
+
y
)
+
g
(
y
)
and
g
(
x
+
f
(
y
))
=
g
(
x
+
y
)
+
f
(
y
)
<I_1DI_2 = 90^o for incenters of ABP, ACP, D touchpoint of incircle with BC
Let
A
B
C
ABC
A
BC
be a triangle and
P
P
P
be any point on the side
B
C
BC
BC
. Let
I
1
I_1
I
1
,
I
2
I_2
I
2
be the incenters of triangles
A
B
P
ABP
A
BP
and
A
C
P
ACP
A
CP
, respectively. If
D
D
D
is the point of tangency of the incircle of
A
B
C
ABC
A
BC
with the side
B
C
BC
BC
, prove that
∠
I
1
D
I
2
=
9
0
o
\angle I_1DI_2 = 90^o
∠
I
1
D
I
2
=
9
0
o
.