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National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2012 Puerto Rico Team Selection Test
2012 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(6)
6
1
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Puerto Rico 2012 TST Problem 6
The increasing sequence
1
;
3
;
4
;
9
;
10
;
12
;
13
;
27
;
28
;
30
;
31
,
…
1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots
1
;
3
;
4
;
9
;
10
;
12
;
13
;
27
;
28
;
30
;
31
,
…
is formed with positive integers which are powers of
3
3
3
or sums of different powers of
3
3
3
. Which number is in the
10
0
t
h
100^{th}
10
0
t
h
position?
5
1
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Puerto Rico 2012 TST Problem 5
A point
P
P
P
is outside of a circle and the distance to the center is
13
13
13
. A secant line from
P
P
P
meets the circle at
Q
Q
Q
and
R
R
R
so that the exterior segment of the secant,
P
Q
PQ
PQ
, is
9
9
9
and
Q
R
QR
QR
is
7
7
7
. Find the radius of the circle.
4
1
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Puerto Rico 2012 TST Problem 4
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be digits such that
d
>
c
>
b
>
a
≥
0
d > c > b > a \geq 0
d
>
c
>
b
>
a
≥
0
. How many numbers of the form
1
a
1
b
1
c
1
d
1
1a1b1c1d1
1
a
1
b
1
c
1
d
1
are multiples of
33
33
33
?
3
1
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Puerto Rico 2012 TST Problem 3
A
B
C
ABC
A
BC
is a triangle that is inscribed in a circle. The angle bisectors of
A
,
B
,
C
A, B, C
A
,
B
,
C
meet the circle at
D
,
E
,
F
D, E, F
D
,
E
,
F
, respectively. Show that
A
D
AD
A
D
is perpendicular to
E
F
EF
EF
.
2
1
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Puerto Rico 2012 TST Problem 2
A cone is constructed with a semicircular piece of paper, with radius 10. Find the height of the cone.
1
1
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Puerto Rico 2012 TST Problem 1
Let
x
,
y
x, y
x
,
y
and
z
z
z
be consecutive integers such that
1
x
+
1
y
+
1
z
>
1
45
.
\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.
x
1
+
y
1
+
z
1
>
45
1
.
Find the maximum value of
x
+
y
+
z
x + y + z
x
+
y
+
z
.