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Problems
Contests
National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2019 Puerto Rico Team Selection Test
2019 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(6)
6
1
Hide problems
max of distances, sequences of figures by pyramids , triangles of midpoints
Starting from a pyramid
T
0
T_0
T
0
whose edges are all of length
2019
2019
2019
, we construct the Figure
T
1
T_1
T
1
when considering the triangles formed by the midpoints of the edges of each face of
T
0
T_0
T
0
, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure
T
2
T_2
T
2
is constructed by applying the same process from
T
1
T_1
T
1
on each triangular face resulting from
T
1
T_1
T
1
, and so on for
T
3
,
T
4
,
.
.
.
T_3, T_4, ...
T
3
,
T
4
,
...
Let
D
0
=
max
{
d
(
x
,
y
)
}
D_0= \max \{d(x,y)\}
D
0
=
max
{
d
(
x
,
y
)}
, where
x
x
x
and
y
y
y
are vertices of
T
0
T_0
T
0
and
d
(
x
,
y
)
d(x,y)
d
(
x
,
y
)
is the distance between
x
x
x
and
y
y
y
. Then we define
D
n
+
1
=
max
{
d
(
x
,
y
)
∣
d
(
x
,
y
)
∉
{
D
0
,
D
1
,
.
.
.
,
D
n
}
D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}
D
n
+
1
=
max
{
d
(
x
,
y
)
∣
d
(
x
,
y
)
∈
/
{
D
0
,
D
1
,
...
,
D
n
}
, where
x
,
y
x, y
x
,
y
are vertices of
T
n
+
1
T_{n+1}
T
n
+
1
. Find the value of
D
n
D_n
D
n
for all
n
n
n
.
5
1
Hide problems
2018 blacks + 1 white magic pearls in Gandalf's necklace Puerto Rico TST 2019.5
The wizard Gandalf has a necklace that is shaped like a row of magic pearls. The necklace has
2019
2019
2019
pearls,
2018
2018
2018
are black and the last one is white. Everytime that the magician Gandalf touches the necklace, the following occurs: the pearl in position
i
i
i
is move to position
i
−
1
i-1
i
−
1
, for
1
<
i
<
2020
1 <i <2020
1
<
i
<
2020
, furthermore the pearl in position
1
1
1
moves to position
2019
2019
2019
. But something else happens, if the pearl in position
1
1
1
now is white, then the last pearl turns white without the need for Gandalf to touch the necklace again. How many times does Gandalf have to touch the necklace to be sure that all pearls are white?
4
1
Hide problems
computational, angle bisector in rectangle Puerto Rico TST 2019.4
Rectangle
A
B
C
D
ABCD
A
BC
D
has sides
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
2
BC = 2
BC
=
2
. Point
P
P
P
lies on side
A
B
AB
A
B
is such that the bisector of the angle
C
D
P
CDP
C
D
P
passes through the midpoint
M
M
M
of
B
C
BC
BC
. Find
B
P
BP
BP
.
3
1
Hide problems
max of a^3b + b^3a if a,b>=0, a+b=3 Puerto Rico TST 2019.3
Find the largest value that the expression can take
a
3
b
+
b
3
a
a^3b + b^3a
a
3
b
+
b
3
a
where
a
,
b
a, b
a
,
b
are non-negative real numbers, with
a
+
b
=
3
a + b = 3
a
+
b
=
3
.
2
1
Hide problems
perpendicularity, square, midpoints, MC = KD Puerto Rico TST 2019.2
Let
A
B
C
D
ABCD
A
BC
D
be a square. Let
M
M
M
and
K
K
K
be points on segments
B
C
BC
BC
and
C
D
CD
C
D
respectively, such that
M
C
=
K
D
MC = KD
MC
=
KD
. Let
P
P
P
be the intersection of the segments
M
D
MD
M
D
and
B
K
BK
B
K
. Prove that
A
P
AP
A
P
is perpendicular to
M
K
MK
M
K
.
1
1
Hide problems
square divides into 25 unit squares Puerto Rico TST 2019.1
A square is divided into
25
25
25
unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?