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Contests
National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2021 Puerto Rico Team Selection Test
2021 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(6)
6
1
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max no allied of 2021, product of primes Puerto Rico TST 2021.6
Two positive integers
n
,
m
≥
2
n,m\ge 2
n
,
m
≥
2
are called allies if when written as a product of primes (not necessarily different):
n
=
p
1
p
2
.
.
.
p
s
n=p_1p_2...p_s
n
=
p
1
p
2
...
p
s
and
m
=
q
1
q
2
.
.
.
q
t
m=q_1q_2...q_t
m
=
q
1
q
2
...
q
t
, turns out that:
p
1
+
p
2
+
.
.
.
+
p
s
=
q
1
+
q
2
+
.
.
.
+
q
t
p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t
p
1
+
p
2
+
...
+
p
s
=
q
1
+
q
2
+
...
+
q
t
(a) Show that the biggest ally of any positive integer has to have only
2
2
2
and
3
3
3
in its prime factorization. (b) Find the biggest number which is allied of
2021
2021
2021
.
5
1
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3 tangent circles in pairs, tangent int. to a 4th Puerto Rico TST 2021.5
Circle
o
o
o
contains the circles
m
m
m
,
p
p
p
and
r
r
r
, such that they are tangent to
o
o
o
internally and any two of them are tangent between themselves. The radii of the circles
m
m
m
and
p
p
p
are equal to
x
x
x
. The circle
r
r
r
has radius
1
1
1
and passes through the center of the circle
o
o
o
. Find the value of
x
x
x
.
4
1
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4-digit no wanted, result in multiple of 30 Puerto Rico TST 2021.4
How many numbers
a
b
c
d
‾
\overline{abcd}
ab
c
d
with different digits satisfy the following property: if we replace the largest digit with the digit
1
1
1
results in a multiple of
30
30
30
?
3
1
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M coins on a n x n board Puerto Rico TST 2021.3
Coins are placed in some squares on a
n
×
n
n\times n
n
×
n
board. Each coin can be moved towards the square symmetrical with respect to either of the two diagonals, as long as that square is empty. The initial coin setup is said to be good , if any coin can make the first move. (a) Determine the maximum number of coins
M
M
M
that can be placed on the
n
×
n
n\times n
n
×
n
board, such that the configuration is good. (b) Calculate the total number of good configurations that have exactly
M
M
M
coins.
2
1
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<IMB=15^o wanted, incenter in 90-60-30 Puerto Rico TST 2021.2
Let
A
B
C
ABC
A
BC
be a right triangle with right angle at
B
B
B
and
∠
C
=
3
0
o
\angle C=30^o
∠
C
=
3
0
o
. If
M
M
M
is midpoint of the hypotenuse and
I
I
I
the incenter of the triangle, show that
∠
I
M
B
=
1
5
o
\angle IMB=15^o
∠
I
MB
=
1
5
o
.
1
1
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winning strategy, first arrives 2021 wins Puerto Rico TST 2021.1
Ana and Beto are playing a game. Ana writes a whole number on the board. Beto then has the right to erase the number and add
2
2
2
to it, or erase the number and subtract
3
3
3
, as many times as he wants. Beto wins if he can get
2021
2021
2021
after a finite number of stages; otherwise, Ana wins. Which player has a winning strategy?