MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Mathematicians for Fun Olympiad
2023 OMpD
2023 OMpD
Part of
Mathematicians for Fun Olympiad
Subcontests
(4)
4
3
Hide problems
x^n+y^n and x^m+y^m integer implies x+y integer
Are there integers
m
,
n
≥
2
m, n \geq 2
m
,
n
≥
2
such that the following property is always true?
‘
‘
For any real numbers
x
,
y
,
if
x
m
+
y
m
and
x
n
+
y
n
are integers, then
x
+
y
is an integer
"
.
``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".
‘‘
For any real numbers
x
,
y
,
if
x
m
+
y
m
and
x
n
+
y
n
are integers, then
x
+
y
is an integer
".
Cute geometry problem
Let
A
B
C
ABC
A
BC
be a scalene acute triangle with circumcenter
O
O
O
. Let
K
K
K
be a point on the side
B
C
‾
\overline{BC}
BC
. Define
M
M
M
as the second intersection of
O
K
↔
\overleftrightarrow{OK}
O
K
with the circumcircle of
B
O
C
BOC
BOC
. Let
L
L
L
be the reflection of
K
K
K
by
A
C
↔
\overleftrightarrow{AC}
A
C
. Show that the circumcircles of the triangles
L
C
M
LCM
L
CM
and
A
B
C
ABC
A
BC
are tangent if, and only if,
A
K
‾
⊥
B
C
‾
\overline{AK} \perp \overline{BC}
A
K
⊥
BC
.
Integral inequality
Let
n
≥
0
n \geq 0
n
≥
0
be an integer and
f
:
[
0
,
1
]
→
R
f: [0, 1] \rightarrow \mathbb{R}
f
:
[
0
,
1
]
→
R
an integrable function such that:
∫
0
1
f
(
x
)
d
x
=
∫
0
1
x
f
(
x
)
d
x
=
∫
0
1
x
2
f
(
x
)
d
x
=
…
=
∫
0
1
x
n
f
(
x
)
d
x
=
1
\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1
∫
0
1
f
(
x
)
d
x
=
∫
0
1
x
f
(
x
)
d
x
=
∫
0
1
x
2
f
(
x
)
d
x
=
…
=
∫
0
1
x
n
f
(
x
)
d
x
=
1
Prove that:
∫
0
1
f
(
x
)
2
d
x
≥
(
n
+
1
)
2
\int_0^1f(x)^2dx \geq (n+1)^2
∫
0
1
f
(
x
)
2
d
x
≥
(
n
+
1
)
2
3
3
Hide problems
Funny game between Humberto and Luciano
Humberto and Luciano use the break between classes to have fun with the following game: Humberto writes a list of distinct positive integers on a green sheet of paper and hands it to Luciano. Luciano then writes on a board all the possible sums, without repetitions, of one or more different numbers written on the green sheet. For example, if Humberto writes
1
1
1
,
3
3
3
and
4
4
4
on the green sheet, Luciano will write
1
1
1
,
3
3
3
,
4
4
4
,
5
5
5
,
7
7
7
and
8
8
8
on the board.(a) Let
n
n
n
be a positive integer. Determine all positive integers
k
k
k
such that Humberto can write a list of
n
n
n
numbers on the green sheet in order to guarantee that Luciano will write exactly
k
k
k
numbers on the board.(b) Luciano now decides to write a list of
m
m
m
distinct positive integers on a yellow sheet of paper. Determine the smallest positive integer
m
m
m
such that it is possible for Luciano to write this list so that, for any list that Humberto writes on the green sheet, with a maximum of
2023
2023
2023
numbers, not all the numbers on the yellow sheet will be written on the board.
a sequence and its phi functions are all consecutive
For each positive integer
x
x
x
, let
φ
(
x
)
\varphi(x)
φ
(
x
)
be the number of integers
1
≤
k
≤
x
1 \leq k \leq x
1
≤
k
≤
x
that do not have prime factors in common with
x
x
x
. Determine all positive integers
n
n
n
such that there are distinct positive integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
so that the set:
S
=
{
a
1
,
a
2
,
…
,
a
n
,
φ
(
a
1
)
,
φ
(
a
2
)
,
…
,
φ
(
a
n
)
}
S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}
S
=
{
a
1
,
a
2
,
…
,
a
n
,
φ
(
a
1
)
,
φ
(
a
2
)
,
…
,
φ
(
a
n
)}
Have exactly
2
n
2n
2
n
consecutive integers (in some order).
Hamiltonian cycle in distance of sets graph
Let
m
m
m
and
n
n
n
be positive integers integers such that
2
m
+
1
<
n
2m + 1 < n
2
m
+
1
<
n
, and let
S
S
S
be the set of the
2
n
2^n
2
n
subsets of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
. Prove that we can place the elements of
S
S
S
on a circle, so that for any two adjacent elements
A
A
A
and
B
B
B
, the set
A
Δ
B
A \Delta B
A
Δ
B
has exactly
2
m
+
1
2m + 1
2
m
+
1
elements.Note:
A
Δ
B
=
(
A
∪
B
)
−
(
A
∩
B
)
A \Delta B = (A \cup B) - (A \cap B)
A
Δ
B
=
(
A
∪
B
)
−
(
A
∩
B
)
is the set of elements that are exclusively in
A
A
A
or exclusively in
B
B
B
.
2
3
Hide problems
Floor of an AP is always a perfect square
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of real numbers such that
⌊
a
n
+
b
⌋
\lfloor an + b \rfloor
⌊
an
+
b
⌋
is a perfect square, for all positive integer
n
n
n
.
Areas with an inscribed pentagon
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon inscribed in a circle
Γ
\Gamma
Γ
, such that
A
B
=
B
C
=
C
D
AB = BC = CD
A
B
=
BC
=
C
D
. Let
F
F
F
and
G
G
G
be the intersections of
B
E
BE
BE
with
A
C
AC
A
C
and of
C
E
CE
CE
with
B
D
BD
B
D
, respectively. Show that:a)
[
A
B
C
]
=
[
F
B
C
G
]
[ABC] = [FBCG]
[
A
BC
]
=
[
FBCG
]
b)
[
E
F
G
]
[
E
A
D
]
=
B
C
A
D
\frac{[EFG]}{[EAD]} = \frac{BC}{AD}
[
E
A
D
]
[
EFG
]
=
A
D
BC
Note:
[
X
]
[X]
[
X
]
denotes the area of polygon
X
X
X
.
At what point will the ants stop?
Let
C
C
C
be a fixed circle,
u
>
0
u > 0
u
>
0
be a fixed real and let
v
0
,
v
1
,
v
2
,
…
v_0 , v_1 , v_2 , \ldots
v
0
,
v
1
,
v
2
,
…
be a sequence of positive real numbers. Two ants
A
A
A
and
B
B
B
walk around the perimeter of
C
C
C
in opposite directions, starting from the same starting point. Ant
A
A
A
has a constant speed
u
u
u
, while ant
B
B
B
has an initial speed
v
0
v_0
v
0
. For each positive integer
n
n
n
, when the two ants collide for the
n
n
n
−th time, they change the directions in which they walk around the perimeter of
C
C
C
, with ant
A
A
A
remaining at speed
u
u
u
and ant
B
B
B
stops walking at speed
v
n
−
1
v_{n-1}
v
n
−
1
to walk at speed
v
n
v_n
v
n
.(a) If the sequence
{
v
n
}
\{v_n\}
{
v
n
}
is strictly increasing, with
lim
n
→
∞
v
n
=
+
∞
\lim_{n\rightarrow \infty} v_n = +\infty
lim
n
→
∞
v
n
=
+
∞
, prove that there is exactly one point in
C
C
C
that ant
A
A
A
will pass "infinitely" many times.(b) Prove that there is a sequence
{
v
n
}
\{v_n\}
{
v
n
}
with
lim
n
→
∞
v
n
=
+
∞
\lim_{n\rightarrow\infty} v_n = +\infty
lim
n
→
∞
v
n
=
+
∞
, such that ant
A
A
A
will pass "infinitely" many times through all points on the circle
C
C
C
.
1
2
Hide problems
easy functional equation
Determine all functions
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that, for all real numbers
x
x
x
and
y
y
y
,
f
(
x
)
(
x
+
f
(
f
(
y
)
)
)
=
f
(
x
2
)
+
x
f
(
y
)
f(x)(x+f(f(y))) = f(x^2)+xf(y)
f
(
x
)
(
x
+
f
(
f
(
y
)))
=
f
(
x
2
)
+
x
f
(
y
)
Easy tournament problem
Some friends formed
6
6
6
football teams, and decided to hold a tournament where each team faces each other exactly once in a match. In each match, whoever wins gets
3
3
3
points, whoever loses gets no points, and if the two teams draw, each gets
1
1
1
point.At the end of the tournament, it was found that the teams' scores were
10
10
10
,
9
9
9
,
6
6
6
,
6
6
6
,
4
4
4
and
2
2
2
points. Regarding this tournament, answer the following items, justifying your answer in each one.(a) How many matches ended in a draw in the tournament?(b) Determine, for each of the
6
6
6
teams, the number of wins, draws and losses.(c) If we consider only the matches played between the team that scored
9
9
9
points against the two teams that scored
6
6
6
points, and the one played between the two teams that scored
6
6
6
points, explain why among these three matches, there are at least
2
2
2
draws.