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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2021 Chile National Olympiad
2021 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(4)
4
1
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A,B,C,D,X concyclic if ABCD cyclic, PC = AD
Consider quadrilateral
A
B
C
D
ABCD
A
BC
D
with
∣
D
C
∣
>
∣
A
D
∣
|DC| > |AD|
∣
D
C
∣
>
∣
A
D
∣
. Let
P
P
P
be a point on
D
C
DC
D
C
such that
P
C
=
A
D
PC = AD
PC
=
A
D
and let
Q
Q
Q
be the midpoint of
D
P
DP
D
P
. Let
L
1
L_1
L
1
be the line perpendicular on
D
C
DC
D
C
passing through
Q
Q
Q
and let
L
2
L_2
L
2
be the bisector of the angle
∠
A
B
C
\angle ABC
∠
A
BC
. Let us call
X
=
L
1
∩
L
2
X = L_1 \cap L_2
X
=
L
1
∩
L
2
. Show that if quadrilateral is cyclic then
X
X
X
lies on the circumcircle of
A
B
C
D
.
ABCD.
A
BC
D
.
https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png
3
1
Hide problems
4p(x^2) = 4(p(x))^2 + 4p(x)- 1
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
with real coefficients that satisfy
4
p
(
x
2
)
=
4
(
p
(
x
)
)
2
+
4
p
(
x
)
−
1
4p(x^2) = 4(p(x))^2 + 4p(x)- 1
4
p
(
x
2
)
=
4
(
p
(
x
)
)
2
+
4
p
(
x
)
−
1
1
1
Hide problems
last digit of a_{2021} if a_n = 7^{a_{n-1}}
Consider the sequence of numbers defined by
a
1
=
7
a_1 = 7
a
1
=
7
,
a
2
=
7
7
a_2 = 7^7
a
2
=
7
7
,
.
.
.
...
...
,
a
n
=
7
a
n
−
1
a_n = 7^{a_{n-1}}
a
n
=
7
a
n
−
1
for
n
≥
2
n \ge 2
n
≥
2
. Determine the last digit of the decimal representation of
a
2021
a_{2021}
a
2021
.
2
1
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array of the digits 1,2,..., 9 in the shape of an X J
A design
X
X
X
is an array of the digits
1
,
2
,
.
.
.
,
9
1,2,..., 9
1
,
2
,
...
,
9
in the shape of an
X
X
X
, for example, https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png We will say that a design
X
X
X
is balanced if the sum of the numbers of each of the diagonals match. Determine the number of designs
X
X
X
that are balanced.