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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2021 Brazil National Olympiad
2021 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(9)
9
1
Hide problems
Floor of the multiples of alpha
Let
α
≥
1
\alpha\geq 1
α
≥
1
be a real number. Define the set
A
(
α
)
=
{
⌊
α
⌋
,
⌊
2
α
⌋
,
⌊
3
α
⌋
,
…
}
A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}
A
(
α
)
=
{⌊
α
⌋
,
⌊
2
α
⌋
,
⌊
3
α
⌋
,
…
}
Suppose that all the positive integers that does not belong to the
A
(
α
)
A(\alpha)
A
(
α
)
are exactly the positive integers that have the same remainder
r
r
r
in the division by
2021
2021
2021
with
0
≤
r
<
2021
0\leq r<2021
0
≤
r
<
2021
. Determine all the possible values of
α
\alpha
α
.
8
1
Hide problems
Mutual divisibility
A triple of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is brazilian if
a
∣
b
c
+
1
a|bc+1
a
∣
b
c
+
1
b
∣
a
c
+
1
b|ac+1
b
∣
a
c
+
1
c
∣
a
b
+
1
c|ab+1
c
∣
ab
+
1
Determine all the brazilian triples.
7
1
Hide problems
Tangent Tangents
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
=
9
0
∘
\angle ABC=90^{\circ}
∠
A
BC
=
9
0
∘
. The square
B
D
E
F
BDEF
B
D
EF
is inscribed in
△
A
B
C
\triangle ABC
△
A
BC
, such that
D
,
E
,
F
D,E,F
D
,
E
,
F
are in the sides
A
B
,
C
A
,
B
C
AB,CA,BC
A
B
,
C
A
,
BC
respectively. The inradius of
△
E
F
C
\triangle EFC
△
EFC
and
△
E
D
A
\triangle EDA
△
E
D
A
are
c
c
c
and
b
b
b
, respectively. Four circles
ω
1
,
ω
2
,
ω
3
,
ω
4
\omega_1,\omega_2,\omega_3,\omega_4
ω
1
,
ω
2
,
ω
3
,
ω
4
are drawn inside the square
B
D
E
F
BDEF
B
D
EF
, such that the radius of
ω
1
\omega_1
ω
1
and
ω
3
\omega_3
ω
3
are both equal to
b
b
b
and the radius of
ω
2
\omega_2
ω
2
and
ω
4
\omega_4
ω
4
are both equal to
a
a
a
. The circle
ω
1
\omega_1
ω
1
is tangent to
E
D
ED
E
D
, the circle
ω
3
\omega_3
ω
3
is tangent to
B
F
BF
BF
,
ω
2
\omega_2
ω
2
is tangent to
E
F
EF
EF
and
ω
4
\omega_4
ω
4
is tangent to
B
D
BD
B
D
, each one of these circles are tangent to the two closest circles and the circles
ω
1
\omega_1
ω
1
and
ω
3
\omega_3
ω
3
are tangents. Determine the ratio
c
a
\frac{c}{a}
a
c
.
6
2
Hide problems
Circle tangent to n circles derived from a bicentric n-gon
Let
n
≥
5
n \geq 5
n
≥
5
be integer. The convex polygon
P
=
A
1
A
2
…
A
n
P = A_{1} A_{2} \ldots A_{n}
P
=
A
1
A
2
…
A
n
is bicentric, that is, it has an inscribed and circumscribed circle. Set
A
i
+
n
=
A
i
A_{i+n}=A_{i}
A
i
+
n
=
A
i
to every integer
i
i
i
(that is, all indices are taken modulo
n
n
n
). Suppose that for all
i
,
1
≤
i
≤
n
i, 1 \leq i \leq n
i
,
1
≤
i
≤
n
, the rays
A
i
−
1
A
i
A_{i-1} A_{i}
A
i
−
1
A
i
and
A
i
+
2
A
i
+
1
A_{i+2} A_{i+1}
A
i
+
2
A
i
+
1
meet at the point
B
i
B_{i}
B
i
. Let
ω
i
\omega_{i}
ω
i
be the circumcircle of
B
i
A
i
A
i
+
1
B_{i} A_{i} A_{i+1}
B
i
A
i
A
i
+
1
. Prove that there is a circle tangent to all
n
n
n
circles
ω
i
\omega_{i}
ω
i
,
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
.
Brazil FC and Graphs
In a football championship with
2021
2021
2021
teams, each team play with another exactly once. The score of the match(es) is three points to the winner, one point to both players if the match end in draw(tie) and zero point to the loser. The final of the tournament will be played by the two highest score teams. Brazil Football Club won the first match, and it has the advantage if in the final score it draws with any other team. Determine the least score such that Brazil Football Club has a chance to play the final match.
5
2
Hide problems
Equation on non-negative integers
Find all triples of non-negative integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
2
+
b
2
+
c
2
=
a
b
c
+
1.
a^{2}+b^{2}+c^{2} = a b c+1.
a
2
+
b
2
+
c
2
=
ab
c
+
1.
Concurrency of simso...
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Let
A
1
A_1
A
1
be the midpoint of the arc
B
C
BC
BC
which contain the point
A
A
A
. Let
A
2
A_2
A
2
and
A
3
A_3
A
3
be the foot(s) of the perpendicular(s) of the point
A
1
A_1
A
1
to the lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Define
B
2
,
B
3
,
C
2
,
C
3
B_2,B_3,C_2,C_3
B
2
,
B
3
,
C
2
,
C
3
analogously. a) Prove that the line
A
2
A
3
A_2A_3
A
2
A
3
cuts
B
C
BC
BC
in the midpoint. b) Prove that the lines
A
2
A
3
,
B
2
B
3
A_2A_3,B_2B_3
A
2
A
3
,
B
2
B
3
and
C
2
C
3
C_2C_3
C
2
C
3
are concurrents.
4
2
Hide problems
Smallest real in a framed set
A set
A
A
A
of real numbers is framed when it is bounded and, for all
a
,
b
∈
A
a, b \in A
a
,
b
∈
A
, not necessarily distinct,
(
a
−
b
)
2
∈
A
(a-b)^{2} \in A
(
a
−
b
)
2
∈
A
. What is the smallest real number that belongs to some framed set?
d(d(d(d(.....)
Let
d
(
n
)
d(n)
d
(
n
)
be the quantity of positive divisors of
n
n
n
, for example
d
(
1
)
=
1
,
d
(
2
)
=
2
,
d
(
10
)
=
4
d(1)=1,d(2)=2,d(10)=4
d
(
1
)
=
1
,
d
(
2
)
=
2
,
d
(
10
)
=
4
. The size of
n
n
n
is
k
k
k
if
k
k
k
is the least positive integer, such that
d
k
(
n
)
=
2
d^k(n)=2
d
k
(
n
)
=
2
. Note that
d
s
(
n
)
=
d
(
d
s
−
1
(
n
)
)
d^s(n)=d(d^{s-1}(n))
d
s
(
n
)
=
d
(
d
s
−
1
(
n
))
. a) How many numbers in the interval
[
3
,
1000
]
[3,1000]
[
3
,
1000
]
have size
2
2
2
? b) Determine the greatest size of a number in the interval
[
3
,
1000
]
[3,1000]
[
3
,
1000
]
.
3
2
Hide problems
Floor of powers of an irrational equals square minus constant
Find all positive integers
k
k
k
for which there is an irrational
α
>
1
\alpha>1
α
>
1
and a positive integer
N
N
N
such that
⌊
α
n
⌋
\left\lfloor\alpha^{n}\right\rfloor
⌊
α
n
⌋
is a perfect square minus
k
k
k
for every integer
n
n
n
with
n
>
N
n>N
n
>
N
.
Geometry is not real
Let
A
B
C
ABC
A
BC
be a scalene triangle and
ω
\omega
ω
is your incircle. The sides
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
are tangents to
ω
\omega
ω
in
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
D
D
D
is the intersection point of
B
C
BC
BC
with the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. Prove that
∠
B
A
X
=
∠
M
A
C
\angle BAX=\angle MAC
∠
B
A
X
=
∠
M
A
C
if and only if
Y
Z
YZ
Y
Z
passes by the midpoint of
A
D
AD
A
D
.
2
2
Hide problems
Double counting
66
66
66
points are given on a plane; collinearity is allowed. There are exactly
2021
2021
2021
lines passing by at least two of the given points. Determine the greatest number of points in a same line. Give an example.
Largest non-crowded subset of 2 by 3n board
Let
n
n
n
be a positive integer. On a
2
×
3
n
2 \times 3 n
2
×
3
n
board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself).(a) What is the greatest possible number of marked square? (b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.
1
2
Hide problems
Four circumcenters not on a circle.
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral in the plane and let
O
A
,
O
B
,
O
C
O_{A}, O_{B}, O_{C}
O
A
,
O
B
,
O
C
and
O
D
O_{D}
O
D
be the circumcenters of the triangles
B
C
D
,
C
D
A
,
D
A
B
BCD, CDA, DAB
BC
D
,
C
D
A
,
D
A
B
and
A
B
C
ABC
A
BC
, respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.
Change or not change the doors
In a school there are
2021
2021
2021
doors with the numbers
1
,
2
,
…
,
2021
1,2,\dots,2021
1
,
2
,
…
,
2021
. In a day
2021
2021
2021
students play the following game: Initially all the doors are closed, and each student receive a card to define the order, there are exactly
2021
2021
2021
cards. The numbers in the cards are
1
,
2
,
…
,
2020
,
2021
1,2,\dots,2020,2021
1
,
2
,
…
,
2020
,
2021
. The order will be student
1
1
1
first, student
2
2
2
will be the second, and going on. The student
k
k
k
will change the state of the doors
k
,
2
k
,
4
k
,
8
k
,
…
,
2
p
k
k,2k,4k,8k,\dots,2^pk
k
,
2
k
,
4
k
,
8
k
,
…
,
2
p
k
with
2
p
k
≤
2021
≤
2
p
+
1
k
2^pk\leq 2021\leq 2^{p+1}k
2
p
k
≤
2021
≤
2
p
+
1
k
. Change the state is if the door was close, it will be open and vice versa. a) After the round of the student
16
16
16
, determine the configuration of the doors
1
,
2
,
…
,
16
1,2,\dots,16
1
,
2
,
…
,
16
b) After the round of the student
2021
2021
2021
, determine how many doors are closed.