MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile TST Ibero
2023 Chile TST Ibero.
2023 Chile TST Ibero.
Part of
Chile TST Ibero
Subcontests
(4)
3
1
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Looks scare but no
Determine the smallest positive integer
n
n
n
with the following property: for every triple of positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
, with
x
x
x
dividing
y
3
y^3
y
3
,
y
y
y
dividing
z
3
z^3
z
3
, and
z
z
z
dividing
x
3
x^3
x
3
, it also holds that
(
x
y
z
)
(xyz)
(
x
yz
)
divides
(
x
+
y
+
z
)
n
(x + y + z)^n
(
x
+
y
+
z
)
n
.
2
1
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Another trivial question
Consider a function
n
↦
f
(
n
)
n \mapsto f(n)
n
↦
f
(
n
)
that satisfies the following conditions:
f
(
n
)
f(n)
f
(
n
)
is an integer for each
n
n
n
.
f
(
0
)
=
1
f(0) = 1
f
(
0
)
=
1
.
f
(
n
+
1
)
>
f
(
n
)
+
f
(
n
−
1
)
+
⋯
+
f
(
0
)
f(n+1) > f(n) + f(n-1) + \cdots + f(0)
f
(
n
+
1
)
>
f
(
n
)
+
f
(
n
−
1
)
+
⋯
+
f
(
0
)
for each
n
=
0
,
1
,
2
,
…
n = 0, 1, 2, \dots
n
=
0
,
1
,
2
,
…
. Determine the smallest possible value of
f
(
2023
)
f(2023)
f
(
2023
)
.
1
1
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C is a integer??
Given a non-negative integer
n
n
n
, determine the values of
c
c
c
for which the sequence of numbers
a
n
=
4
n
c
+
4
n
−
(
−
1
)
n
5
a_n = 4^n c + \frac{4^n - (-1)^n}{5}
a
n
=
4
n
c
+
5
4
n
−
(
−
1
)
n
contains at least one perfect square.
4
1
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TST IBERO 2023 CHILE
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
and let
ω
\omega
ω
be its circumcircle. Let
M
M
M
denote the midpoint of side
B
C
BC
BC
and
N
N
N
the midpoint of arc
B
C
BC
BC
of
ω
\omega
ω
that contains
A
A
A
. The circumcircle of triangle
A
M
N
AMN
A
MN
intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that
B
P
=
C
Q
BP = CQ
BP
=
CQ
.