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National and Regional Contests
Brazil Contests
Brazil Team Selection Test
2023 Brazil Team Selection Test
2023 Brazil Team Selection Test
Part of
Brazil Team Selection Test
Subcontests
(4)
4
1
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Cute and easy NT problem
Find all positive integers
n
n
n
with the following property: There are only a finite number of positive multiples of
n
n
n
that have exactly
n
n
n
positive divisors.
3
1
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Strange inequality with Max function, AM and GM
Show that for all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
, we have that
a
+
b
+
c
3
−
a
b
c
3
≤
max
{
(
a
−
b
)
2
,
(
b
−
c
)
2
,
(
c
−
a
)
2
}
\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}
3
a
+
b
+
c
−
3
ab
c
≤
max
{(
a
−
b
)
2
,
(
b
−
c
)
2
,
(
c
−
a
)
2
}
2
1
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Just a symmedian problem
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. The tangent to the circumcircle of triangle
B
C
D
BCD
BC
D
at
C
C
C
intersects
A
B
AB
A
B
at
P
P
P
and intersects
A
D
AD
A
D
at
Q
Q
Q
. The tangents to the circumcircle of triangle
A
P
Q
APQ
A
PQ
at
P
P
P
and
Q
Q
Q
meet at
R
R
R
. Show that points
A
A
A
,
C
C
C
, and
R
R
R
are collinear.
1
1
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Students and clubs
In a school there are
1200
1200
1200
students. Each student is part of exactly
k
k
k
clubs. For any
23
23
23
students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of
k
k
k
.