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National and Regional Contests
Brazil Contests
Brazil Team Selection Test
2013 Brazil Team Selection Test
2013 Brazil Team Selection Test
Part of
Brazil Team Selection Test
Subcontests
(3)
1
1
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D in AB such that AB,BC,CA,CD are integers and AD/DB=9/7
Find a triangle
A
B
C
ABC
A
BC
with a point
D
D
D
on side
A
B
AB
A
B
such that the measures of
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
and
C
D
CD
C
D
are all integers and
A
D
D
B
=
9
7
\frac{AD}{DB}=\frac{9}{7}
D
B
A
D
=
7
9
, or prove that such a triangle does not exist.
2
1
Hide problems
<ALK acute wanted, starting with cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a convex cyclic quadrilateral with
A
D
>
B
C
AD > BC
A
D
>
BC
, A
B
B
B
not being diameter and
C
D
C D
C
D
belonging to the smallest arc
A
B
AB
A
B
of the circumcircle. The rays
A
D
AD
A
D
and
B
C
BC
BC
are cut at
K
K
K
, the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are cut at
P
P
P
and the line
K
P
KP
K
P
cuts the side
A
B
AB
A
B
at point
L
L
L
. Prove that angle
∠
A
L
K
\angle ALK
∠
A
L
K
is acute.
4
1
Hide problems
\sqrt{a^2+bc} + \sqrt{b^2+ac} + \sqrt{c^2+ab} <= 3 if a + b + c <= 2.
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be non-negative reals with
a
+
b
+
c
≤
2
a + b + c \le 2
a
+
b
+
c
≤
2
. prove that
b
2
+
a
c
+
a
2
+
b
c
+
c
2
+
a
b
≤
3
\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3
b
2
+
a
c
+
a
2
+
b
c
+
c
2
+
ab
≤
3