MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2015 Brazil National Olympiad
2015 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
4
1
Hide problems
Divisors
Let
n
n
n
be a integer and let
n
=
d
1
>
d
2
>
⋯
>
d
k
=
1
n=d_1>d_2>\cdots>d_k=1
n
=
d
1
>
d
2
>
⋯
>
d
k
=
1
its positive divisors. a) Prove that
d
1
−
d
2
+
d
3
−
⋯
+
(
−
1
)
k
−
1
d
k
=
n
−
1
d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1
d
1
−
d
2
+
d
3
−
⋯
+
(
−
1
)
k
−
1
d
k
=
n
−
1
iff
n
n
n
is prime or
n
=
4
n=4
n
=
4
. b) Determine the three positive integers such that
d
1
−
d
2
+
d
3
−
.
.
.
+
(
−
1
)
k
−
1
d
k
=
n
−
4.
d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.
d
1
−
d
2
+
d
3
−
...
+
(
−
1
)
k
−
1
d
k
=
n
−
4.
5
1
Hide problems
Non integer polynomial f
Is that true that there exist a polynomial
f
(
x
)
f(x)
f
(
x
)
with rational coefficients, not all integers, with degree
n
>
0
n>0
n
>
0
, a polynomial
g
(
x
)
g(x)
g
(
x
)
, with integer coefficients, and a set
S
S
S
with
n
+
1
n+1
n
+
1
integers such that
f
(
t
)
=
g
(
t
)
f(t)=g(t)
f
(
t
)
=
g
(
t
)
for all
t
∈
S
t \in S
t
∈
S
?
3
1
Hide problems
f(n)=f(n-1)+1
Given a natural
n
>
1
n>1
n
>
1
and its prime fatorization
n
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}
n
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
, its false derived is defined by
f
(
n
)
=
α
1
p
1
α
1
−
1
α
2
p
2
α
2
−
1
.
.
.
α
k
p
k
α
k
−
1
.
f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.
f
(
n
)
=
α
1
p
1
α
1
−
1
α
2
p
2
α
2
−
1
...
α
k
p
k
α
k
−
1
.
Prove that there exist infinitely many naturals
n
n
n
such that
f
(
n
)
=
f
(
n
−
1
)
+
1
f(n)=f(n-1)+1
f
(
n
)
=
f
(
n
−
1
)
+
1
.
2
1
Hide problems
Subset with 4n elements
Consider
S
=
{
1
,
2
,
3
,
⋯
,
6
n
}
S=\{1, 2, 3, \cdots, 6n\}
S
=
{
1
,
2
,
3
,
⋯
,
6
n
}
,
n
>
1
n>1
n
>
1
. Find the largest
k
k
k
such that the following statement is true: every subset
A
A
A
of
S
S
S
with
4
n
4n
4
n
elements has at least
k
k
k
pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
,
a
<
b
a<b
a
<
b
and
b
b
b
is divisible by
a
a
a
.
6
1
Hide problems
Circumference with diameter HG
Let
△
A
B
C
\triangle ABC
△
A
BC
be a scalene triangle and
X
X
X
,
Y
Y
Y
and
Z
Z
Z
be points on the lines
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, such that
∡
A
X
B
=
∡
B
Y
C
=
∡
C
Z
A
\measuredangle AXB = \measuredangle BYC = \measuredangle CZA
∡
A
XB
=
∡
B
Y
C
=
∡
CZ
A
. The circumcircles of
B
X
Z
BXZ
BXZ
and
C
X
Y
CXY
CX
Y
intersect at
P
P
P
. Prove that
P
P
P
is on the circumference which diameter has ends in the ortocenter
H
H
H
and in the baricenter
G
G
G
of
△
A
B
C
\triangle ABC
△
A
BC
.
1
1
Hide problems
A, D, N are collinear iff BAC = 45º
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute-scalene triangle, and let
N
N
N
be the center of the circle wich pass trough the feet of altitudes. Let
D
D
D
be the intersection of tangents to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
B
B
B
and
C
C
C
. Prove that
A
A
A
,
D
D
D
and
N
N
N
are collinear iff
∡
B
A
C
=
45
º
\measuredangle BAC = 45º
∡
B
A
C
=
45º
.