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Problems
Contests
National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2023 Puerto Rico Team Selection Test
2023 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(8)
8
1
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computational geo for the finale (Puerto Rico TST 2023.1.8)
Inside a quadrilateral
A
B
C
D
ABCD
A
BC
D
there exists a point
P
P
P
such that
A
P
AP
A
P
is perpendicular to
A
D
AD
A
D
and the line
B
P
BP
BP
is perpendicular to
D
C
DC
D
C
. Besides,
A
B
=
7
AB = 7
A
B
=
7
,
A
P
=
3
AP = 3
A
P
=
3
,
B
P
=
6
BP = 6
BP
=
6
,
A
D
=
5
AD = 5
A
D
=
5
and
C
D
=
10
CD = 10
C
D
=
10
. Calculate the area of the triangle
A
B
C
ABC
A
BC
.
7
1
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2023 wise men on a circle, 2 opinions (Puerto Rico TST 2023.1.7)
2023
2023
2023
wise men are located in a circle. Each of them thinks either that the earth is the center of the universe, or that it is not. Once a minute, all the wise men express their opinion at the same time. Every wise man who is between two wise men with an opinion different from his will change his mind at that moment. The others don't change their minds. The others don't change their minds. Determine the smallest necessary time for all the wise men to have the same opinion, without regardless of initial opinions or your location.
6
1
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integer value of (ab+ 2023 b/4a) (Puerto Rico TST 2023.1.6)
Find all possible integer values of the sum:
a
b
+
2023
×
b
4
×
a
,
\frac{a}{b}+ \frac{2023 \times b}{4 \times a},
b
a
+
4
×
a
2023
×
b
,
where
a
a
a
and
b
b
b
are positive integers with no prime factors in common.
5
1
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6 fruit baskets contain peaches, apples and pears (Puerto Rico TST 2023.1.5)
Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place
31
31
31
fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of
31
31
31
.
4
2
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n = a^2 + b^2 + c^2 + d^2 - Puerto Rico TST 2023.1.4
Find all positive integers
n
n
n
such that:
n
=
a
2
+
b
2
+
c
2
+
d
2
,
n = a^2 + b^2 + c^2 + d^2,
n
=
a
2
+
b
2
+
c
2
+
d
2
,
where
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
are the smallest divisors of
n
n
n
.
3 jumps of a frog on lattice points
A frog started from the origin of the coordinate plane and made
3
3
3
jumps. Each time, the frog jumped a distance of
5
5
5
units and landed on a point with integer coordinates. How many different position possibilities end of the frog there?
3
2
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2023 numbers from {-1,0,12} Puerto Rico TST 2023.1.3
You have a list of
2023
2023
2023
numbers, where each one can be
−
1
-1
−
1
,
0
0
0
,
1
1
1
or
2
2
2
. The sum of all numbers is
19
19
19
and the sum of their squares is
99
99
99
. What are the minimum and maximum values of the sum of the cubes of those
2023
2023
2023
numbers?
p(2023) =? if p(k) =1/(k+1)
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree
2022
2022
2022
such that:
p
(
k
)
=
1
k
+
1
for
k
=
0
,
1
,
.
.
.
,
2022
p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022
p
(
k
)
=
k
+
1
1
for
k
=
0
,
1
,
...
,
2022
Find
p
(
2023
)
p(2023)
p
(
2023
)
.
2
2
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isosceles wanted, semicircle related Puerto Rico TST 2023.1.2
Consider a semicircle with center
M
M
M
and diameter
A
B
AB
A
B
. Let
P
P
P
be a point in the semicircle, different from
A
A
A
and
B
B
B
, and let
Q
Q
Q
be the midpoint of the arc
A
P
AP
A
P
. The line parallel to
Q
P
QP
QP
through
M
M
M
intersects
P
B
PB
PB
at the point
S
S
S
. Prove that the triangle
P
M
S
PMS
PMS
is isosceles.
AP _|_ BP wanted, 2 touchpoints of incircle related
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
and let
D
D
D
and
E
E
E
be the touchpoints of the incircle with sides
B
C
BC
BC
and
A
C
AC
A
C
, respectively. The lines
D
E
DE
D
E
and
B
I
BI
B
I
intersect at point
P
P
P
. Prove that
A
P
AP
A
P
is perpendicular to
B
P
BP
BP
.
1
2
Hide problems
2023-digit palindromes - Puerto Rico TST 2023.1.1
A number is capicua if it is read equally from left to right as it is from right to the left. For example,
23432
23432
23432
and
111111
111111
111111
are capicua numbers. (a) How many
2023
2023
2023
-digit capicua numbers can be formed if you want them to have at least
2022
2022
2022
equal digits? (b) How many
2023
2023
2023
-digit capicua numbers can be formed if you want them to have at least
2021
2021
2021
equal digits?
a! +b! = 2^{c!}
Determine all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
a
!
+
b
!
=
2
c
!
.
a! +b! = 2^{c!} .
a
!
+
b
!
=
2
c
!
.