MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2018 Brazil National Olympiad
2018 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
2
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Points and triangles in the plane
Consider
4
n
4n
4
n
points in the plane, with no three points collinear. Using these points as vertices, we form
(
4
n
3
)
\binom{4n}{3}
(
3
4
n
)
triangles. Show that there exists a point
X
X
X
of the plane that belongs to the interior of at least
2
n
3
2n^3
2
n
3
of these triangles.
Sum of digits
Let
S
(
n
)
S(n)
S
(
n
)
be the sum of digits of
n
n
n
. Determine all the pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers, such that the expression
S
(
a
n
+
b
)
−
S
(
n
)
S(an + b) - S(n)
S
(
an
+
b
)
−
S
(
n
)
has a finite number of values, where
n
n
n
is varying in the positive integers.
5
2
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Multiples of 7 in the sequence 1, 12, 123, ...
Consider the sequence in which
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
a_n
a
n
is obtained by juxtaposing the decimal representation of
n
n
n
at the end of the decimal representation of
a
n
−
1
a_{n-1}
a
n
−
1
. That is,
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
12
a_2 = 12
a
2
=
12
,
a
3
=
123
a_3 = 123
a
3
=
123
,
…
\dots
…
,
a
9
=
123456789
a_9 = 123456789
a
9
=
123456789
,
a
10
=
12345678910
a_{10} = 12345678910
a
10
=
12345678910
and so on. Prove that infinitely many numbers of this sequence are multiples of
7
7
7
.
Operations
One writes, initially, the numbers
1
,
2
,
3
,
…
,
10
1,2,3,\dots,10
1
,
2
,
3
,
…
,
10
in a board. An operation is to delete the numbers
a
,
b
a, b
a
,
b
and write the number
a
+
b
+
a
b
f
(
a
,
b
)
a+b+\frac{ab}{f(a,b)}
a
+
b
+
f
(
a
,
b
)
ab
, where
f
(
a
,
b
)
f(a, b)
f
(
a
,
b
)
is the sum of all numbers in the board excluding
a
a
a
and
b
b
b
, one will make this until remain two numbers
x
,
y
x, y
x
,
y
with
x
≥
y
x\geq y
x
≥
y
. Find the maximum value of
x
x
x
.
4
2
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Maximum value of a cyclic sum
Esmeralda writes
2
n
2n
2
n
real numbers
x
1
,
x
2
,
…
,
x
2
n
x_1, x_2, \dots , x_{2n}
x
1
,
x
2
,
…
,
x
2
n
, all belonging to the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products
p
1
=
x
1
x
2
p_1 = x_1x_2
p
1
=
x
1
x
2
,
p
2
=
x
2
x
3
p_2 = x_2x_3
p
2
=
x
2
x
3
,
…
\dots
…
,
p
2
n
=
x
2
n
x
1
p_{2n} = x_{2n}x_1
p
2
n
=
x
2
n
x
1
. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
Incentrics
a) In a
X
Y
Z
XYZ
X
Y
Z
triangle, the incircle tangents the
X
Y
XY
X
Y
and
X
Z
XZ
XZ
sides at the
T
T
T
and
W
W
W
points, respectively. Prove that:
X
T
=
X
W
=
X
Y
+
X
Z
−
Y
Z
2
XT = XW = \frac {XY + XZ-YZ} {2}
XT
=
X
W
=
2
X
Y
+
XZ
−
Y
Z
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
is the foot of the relative height next to
A
.
A.
A
.
Are
I
I
I
and
J
J
J
the incentives from triangle
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
, respectively. The circles of
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
tangency
A
D
AD
A
D
at points
M
M
M
and
N
N
N
, respectively. Let
P
P
P
be the tangency point of the
B
C
BC
BC
circle with the
A
B
AB
A
B
side. The center circle
A
A
A
and radius
A
P
AP
A
P
intersect the height
D
D
D
at
K
.
K.
K
.
b) Show that the triangles
I
M
K
IMK
I
M
K
and
K
N
J
KNJ
K
N
J
are congruent c) Show that the
I
D
J
K
IDJK
I
D
J
K
quad is inscritibed
3
2
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Game with number theory
Let
k
k
k
,
n
n
n
be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers
1
,
2
…
,
n
1, 2 \dots, n
1
,
2
…
,
n
, with
1
1
1
and
n
n
n
neighbors. It is known that pin
1
1
1
is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer
d
d
d
any and Bernaldo moves the ring
d
d
d
pins clockwise or counterclockwise (positions are considered modulo
n
n
n
, i.e., pins
x
x
x
,
y
y
y
equal if and only if
n
n
n
divides
x
−
y
x-y
x
−
y
). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:Rule 1: Arnaldo chooses a positive integer
d
d
d
and Bernaldo moves the ring
d
d
d
pins clockwise or counterclockwise.Rule 2: Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in
d
d
d
or
k
d
kd
k
d
pins, where
d
d
d
is the size of the last displacement performed.Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of
k
k
k
, the values of
n
n
n
for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.
Concurrency
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumcenter
O
O
O
and orthocenter
H
H
H
. The circle with center
X
a
X_a
X
a
passes in the points
A
A
A
and
H
H
H
and is tangent to the circumcircle of
A
B
C
ABC
A
BC
. Define
X
b
,
X
c
X_b, X_c
X
b
,
X
c
analogously, let
O
a
,
O
b
,
O
c
O_a, O_b, O_c
O
a
,
O
b
,
O
c
the symmetric of
O
O
O
to the sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
, respectively. Prove that the lines
O
a
X
a
,
O
b
X
b
,
O
c
X
c
O_aX_a, O_bX_b, O_cX_c
O
a
X
a
,
O
b
X
b
,
O
c
X
c
are concurrents.
2
2
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Brazilian Mathematical Olympiad N2 (7th-8th grade) Day 1 P2
We say that a quadruple
(
A
,
B
,
C
,
D
)
(A,B,C,D)
(
A
,
B
,
C
,
D
)
is dobarulho when
A
,
B
,
C
A,B,C
A
,
B
,
C
are non-zero algorisms and
D
D
D
is a positive integer such that:
1.
1.
1.
A
≤
8
A \leq 8
A
≤
8
2.
2.
2.
D
>
1
D>1
D
>
1
3.
3.
3.
D
D
D
divides the six numbers
A
B
C
‾
\overline{ABC}
A
BC
,
B
C
A
‾
\overline{BCA}
BC
A
,
C
A
B
‾
\overline{CAB}
C
A
B
,
(
A
+
1
)
C
B
‾
\overline{(A+1)CB}
(
A
+
1
)
CB
,
C
B
(
A
+
1
)
‾
\overline{CB(A+1)}
CB
(
A
+
1
)
,
B
(
A
+
1
)
C
‾
\overline{B(A+1)C}
B
(
A
+
1
)
C
. Find all such quadruples.
Operations on a blackboard
Azambuja writes a rational number
q
q
q
on a blackboard. One operation is to delete
q
q
q
and replace it by
q
+
1
q+1
q
+
1
; or by
q
−
1
q-1
q
−
1
; or by
q
−
1
2
q
−
1
\frac{q-1}{2q-1}
2
q
−
1
q
−
1
if
q
≠
1
2
q \neq \frac{1}{2}
q
=
2
1
. The final goal of Azambuja is to write the number
1
2018
\frac{1}{2018}
2018
1
after performing a finite number of operations. a) Show that if the initial number written is
0
0
0
, then Azambuja cannot reach his goal. b) Find all initial numbers for which Azambuja can achieve his goal.
1
2
Hide problems
Inequality with triangles and squares
We say that a polygon
P
P
P
is inscribed in another polygon
Q
Q
Q
when all vertices of
P
P
P
belong to perimeter of
Q
Q
Q
. We also say in this case that
Q
Q
Q
is circumscribed to
P
P
P
. Given a triangle
T
T
T
, let
l
l
l
be the maximum value of the side of a square inscribed in
T
T
T
and
L
L
L
be the minimum value of the side of a square circumscribed to
T
T
T
. Prove that for every triangle
T
T
T
the inequality
L
/
l
≥
2
L/l \ge 2
L
/
l
≥
2
holds and find all the triangles
T
T
T
for which the equality occurs.
ropeticks
Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number.