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National and Regional Contests
Azerbaijan Contests
Azerbaijan Team Selection Test
2017 Azerbaijan Team Selection Test
2017 Azerbaijan Team Selection Test
Part of
Azerbaijan Team Selection Test
Subcontests
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Circle Passing Through a Fixed Point
Let
A
B
C
ABC
A
BC
be an acute angled triangle. Points
E
E
E
and
F
F
F
are chosen on the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, such that
B
C
2
=
B
A
×
B
F
+
C
E
×
C
A
.
BC^2=BA\times BF+CE\times CA.
B
C
2
=
B
A
×
BF
+
CE
×
C
A
.
Prove that for all such
E
E
E
and
F
F
F
, circumcircle of the triangle
A
E
F
AEF
A
EF
passes through a fixed point different from
A
A
A
.
Sequence with Number Theory
Consider the sequence of rational numbers defined by
x
1
=
4
3
x_1=\frac{4}{3}
x
1
=
3
4
, and
x
n
+
1
=
x
n
2
x
n
2
−
x
n
+
1
x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}
x
n
+
1
=
x
n
2
−
x
n
+
1
x
n
2
. Show that the nu,erator of the lowest term expression of each sum
x
1
+
x
2
+
.
.
.
+
x
k
x_1+x_2+...+x_k
x
1
+
x
2
+
...
+
x
k
is a perfect square.