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National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2020 Puerto Rico Team Selection Test
2020 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(4)
2
1
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diophantine system in a algebra problem Puerto Rico TST 2020.2
The cost of
1000
1000
1000
grams of chocolate is
x
x
x
dollars and the cost of
1000
1000
1000
grams of potatoes is
y
y
y
dollars, the numbers
x
x
x
and
y
y
y
are positive integers and have not more than
2
2
2
digits. Mother said to Maria to buy
200
200
200
grams of chocolate and
1000
1000
1000
grams of potatoes that cost exactly
N
N
N
dollars. Maria got confused and bought
1000
1000
1000
grams of chocolate and
200
200
200
grams of potatoes that cost exactly
M
M
M
dollars (
M
>
N
M >N
M
>
N
). It turned out that the numbers
M
M
M
and
N
N
N
have no more than two digits and are formed of the same digits but in a different order. Find
x
x
x
and
y
y
y
.
4
1
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dissect square m x m into 5 rectangles Puerto Rico TST 2020.4
Determine all integers
m
m
m
, for which it is possible to dissect the square
m
×
m
m\times m
m
×
m
into five rectangles, with the side lengths being the integers
1
,
2
,
…
,
10
1, 2, … ,10
1
,
2
,
…
,
10
in some order.
3
1
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ABC right triangle if CD=BC, AE=2CA, AD=BE Puerto Rico TST 2020.3
The side
B
C
BC
BC
of the triangle
A
B
C
ABC
A
BC
is extended beyond
C
C
C
to
D
D
D
, such that
C
D
=
B
C
CD=BC
C
D
=
BC
. The side
C
A
CA
C
A
is extended beyond
A
A
A
to
E
E
E
, such that
A
E
=
2
C
A
AE=2CA
A
E
=
2
C
A
. Prove that if
A
D
=
B
E
AD=BE
A
D
=
BE
, then the triangle
A
B
C
ABC
A
BC
is right.
1
1
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max hexagon by 10,000 equilaterals Puerto Rico TST 2020.1
We have
10
,
000
10,000
10
,
000
identical equilateral triangles. Consider the largest regular hexagon that can be formed with these triangles without overlapping. How many triangles will not be used?