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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil Team Selection Test
2001 Brazil Team Selection Test
2001 Brazil Team Selection Test
Part of
Brazil Team Selection Test
Subcontests
(4)
Problem 2
2
Hide problems
n is power of 2
Let
f
(
n
)
f(n)
f
(
n
)
denote the least positive integer
k
k
k
such that
1
+
2
+
⋯
+
k
1+2+\cdots+k
1
+
2
+
⋯
+
k
is divisible by
n
n
n
. Show that
f
(
n
)
=
2
n
−
1
f(n)=2n-1
f
(
n
)
=
2
n
−
1
if and only if
n
n
n
is a power of
2
2
2
.
minimization in a floor sequence
A set
S
S
S
consists of
k
k
k
sequences of
0
,
1
,
2
0,1,2
0
,
1
,
2
of length
n
n
n
. For any two sequences
(
a
i
)
,
(
b
i
)
∈
S
(a_i),(b_i)\in S
(
a
i
)
,
(
b
i
)
∈
S
we can construct a new sequence
(
c
i
)
(c_i)
(
c
i
)
such that
c
i
=
⌊
a
i
+
b
i
+
1
2
⌋
c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor
c
i
=
⌊
2
a
i
+
b
i
+
1
⌋
and include it in
S
S
S
. Assume that after performing finitely many such operations we obtain all the
3
n
3n
3
n
sequences of
0
,
1
,
2
0,1,2
0
,
1
,
2
of length
n
n
n
. Find the least possible value of
k
k
k
.
Problem 1
2
Hide problems
strange function
Find all functions
f
f
f
defined on real numbers and taking values in the set of real numbers such that
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
≥
f
(
x
+
2
y
+
3
z
)
f(x+y)+f(y+z)+f(z+x) \geq f(x+2y+3z)
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
≥
f
(
x
+
2
y
+
3
z
)
for all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
.There is an infinity of such functions. Every function with the property that
3
inf
f
≥
sup
f
3 \inf f \geq \sup f
3
in
f
f
≥
sup
f
is a good one. I wonder if there is a way to find all the solutions. It seems very strange.
good
given that p,q are two polynomials such that each one has at least one root and
p
(
1
+
x
+
q
(
x
)
2
)
=
q
(
1
+
x
+
p
(
x
)
2
)
p(1+x+q(x)^2)=q(1+x+p(x)^2)
p
(
1
+
x
+
q
(
x
)
2
)
=
q
(
1
+
x
+
p
(
x
)
2
)
then prove that p=q
Problem 3
2
Hide problems
similar to USAMO 2018/6
For which positive integers
n
n
n
is there a permutation
(
x
1
,
x
2
,
…
,
x
n
)
(x_1,x_2,\ldots,x_n)
(
x
1
,
x
2
,
…
,
x
n
)
of
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
such that all the differences
∣
x
k
−
k
∣
|x_k-k|
∣
x
k
−
k
∣
,
k
=
1
,
2
,
…
,
n
k = 1,2,\ldots,n
k
=
1
,
2
,
…
,
n
, are distinct?
Internal and external bisectors of A meet BC at D,E
In a triangle
A
B
C
,
ABC,
A
BC
,
the internal and external bisectors of the angle
A
A
A
intersect the line
B
C
BC
BC
at
D
D
D
and
E
E
E
respectively. The line
A
C
AC
A
C
meets the circle with diameter
D
E
DE
D
E
again at
F
.
F.
F
.
The tangent line to the circle
A
B
F
ABF
A
BF
at
A
A
A
meets the circle with diameter
D
E
DE
D
E
again at
G
.
G.
G
.
Show that
A
F
=
A
G
.
AF = AG.
A
F
=
A
G
.
Problem 4
2
Hide problems
A, P, Q, O are concyclic if BP : PQ : QC = b : a : c
Let
A
B
C
ABC
A
BC
be a triangle with circumcenter
O
O
O
. Let
P
P
P
and
Q
Q
Q
be points on the segments
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
P
:
P
Q
:
Q
C
=
A
C
:
C
B
:
B
A
BP : PQ : QC = AC : CB : BA
BP
:
PQ
:
QC
=
A
C
:
CB
:
B
A
. Prove that the points
A
A
A
,
P
P
P
,
Q
Q
Q
and
O
O
O
lie on one circle. Alternative formulation. Let
O
O
O
be the center of the circumcircle of a triangle
A
B
C
ABC
A
BC
. If
P
P
P
and
Q
Q
Q
are points on the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, satisfying
B
P
P
Q
=
C
A
B
C
\frac{BP}{PQ}=\frac{CA}{BC}
PQ
BP
=
BC
C
A
and
C
Q
P
Q
=
A
B
B
C
\frac{CQ}{PQ}=\frac{AB}{BC}
PQ
CQ
=
BC
A
B
, then show that the points
A
A
A
,
P
P
P
,
Q
Q
Q
and
O
O
O
lie on one circle.
modular congruence for sequence
Prove that for all integers
n
≥
3
n\ge3
n
≥
3
there exists a set
A
n
=
{
a
1
,
a
2
,
…
,
a
n
}
A_n=\{a_1,a_2,\ldots,a_n\}
A
n
=
{
a
1
,
a
2
,
…
,
a
n
}
of
n
n
n
distinct natural numbers such that, for each
i
=
1
,
2
,
…
,
n
i=1,2,\ldots,n
i
=
1
,
2
,
…
,
n
,
∏
1
≤
k
≤
n
k
≠
i
a
k
≡
1
(
m
o
d
a
i
)
.
\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.
1
≤
k
≤
n
k
=
i
∏
a
k
≡
1
(
mod
a
i
)
.