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Problems
Contests
National and Regional Contests
Bolivia Contests
Bolivian IMO TST
2022 Bolivia IMO TST
2022 Bolivia IMO TST
Part of
Bolivian IMO TST
Subcontests
(2)
P3
1
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GEOMETRY!!! (but sadly it is a P1 :c)
On
△
A
B
C
\triangle ABC
△
A
BC
, let
M
M
M
the midpoint of
A
B
AB
A
B
and
N
N
N
the midpoint of
C
M
CM
CM
. Let
X
X
X
a point such that
∠
X
M
C
=
∠
M
B
C
\angle XMC=\angle MBC
∠
XMC
=
∠
MBC
and
∠
X
C
M
=
∠
M
C
B
\angle XCM=\angle MCB
∠
XCM
=
∠
MCB
with
X
,
B
X,B
X
,
B
in opposite sides of line
C
M
CM
CM
. Let
Ω
\Omega
Ω
the circumcircle of triangle
△
A
M
X
\triangle AMX
△
A
MX
a) Show that
C
M
CM
CM
is tangent to
Ω
\Omega
Ω
b) Show that the lines
N
X
NX
NX
and
A
C
AC
A
C
meet at
Ω
\Omega
Ω
P1
1
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Friendly problem + a monster at TST
Find all possible values of
1
x
+
1
y
\frac{1}{x}+\frac{1}{y}
x
1
+
y
1
, if
x
,
y
x,y
x
,
y
are real numbers not equal to
0
0
0
that satisfy
x
3
+
y
3
+
3
x
2
y
2
=
x
3
y
3
x^3+y^3+3x^2y^2=x^3y^3
x
3
+
y
3
+
3
x
2
y
2
=
x
3
y
3