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National and Regional Contests
Estonia Contests
SAFEST (South Africa - Estonia) Olympiad
2021 SAFEST Olympiad
2021 SAFEST Olympiad
Part of
SAFEST (South Africa - Estonia) Olympiad
Subcontests
(2)
5
1
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find polynomials with real coefficients such zP(z) = 1 iff P(z-1)P(z + 1) = 0
Find all polynomials
P
P
P
with real coefficients having no repeated roots, such that for any complex number
z
z
z
, the equation
z
P
(
z
)
=
1
zP(z) = 1
z
P
(
z
)
=
1
holds if and only if
P
(
z
−
1
)
P
(
z
+
1
)
=
0
P(z-1)P(z + 1) = 0
P
(
z
−
1
)
P
(
z
+
1
)
=
0
.Remark: Remember that the roots of a polynomial are not necessarily real numbers.
4
1
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<KCD = <DAC if DB = DC
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
>
A
C
AB > AC
A
B
>
A
C
. Let
D
D
D
be a point on the side
A
B
AB
A
B
such that
D
B
=
D
C
DB = DC
D
B
=
D
C
and let
M
M
M
be the midpoint of
A
C
AC
A
C
. The line parallel to
B
C
BC
BC
passing through
D
D
D
intersects the line
B
M
BM
BM
in
K
K
K
. Show that
∠
K
C
D
=
∠
D
A
C
.
\angle KCD = \angle DAC.
∠
K
C
D
=
∠
D
A
C
.