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National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2010 Puerto Rico Team Selection Test
2010 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(6)
1
2
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computational, 2 circles - Puerto Rico TST 2010.1
The circles in the figure have their centers at
C
C
C
and
D
D
D
and intersect at
A
A
A
and
B
B
B
. Let
∠
A
C
B
=
60
\angle ACB =60
∠
A
CB
=
60
,
∠
A
D
B
=
9
0
o
\angle ADB =90^o
∠
A
D
B
=
9
0
o
and
D
A
=
1
DA = 1
D
A
=
1
. Find the length of
C
A
CA
C
A
. https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png
2 player game with 3 dice Puerto Rico Ibero TST 2010.1
Maria and Luis play the following game: Maria throws three dice and Luis can select some of them (possibly none) and turn them changing their value for the value in the opposite face of each selected die. Prove that Luis can always play in such a way that the sum of the upper faces of the dice after the change is a multiple of
4
4
4
. Note: The game is played with normal dice, that is, the sum of opposite faces is
7
7
7
.
6
1
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find r if r (ab + bc + ca) + (3 r) (1/a+1/b+1/c) >=9 for a,b,c>0
Find all values of
r
r
r
such that the inequality
r
(
a
b
+
b
c
+
c
a
)
+
(
3
−
r
)
(
1
a
+
1
b
+
1
c
)
≥
9
r (ab + bc + ca) + (3- r) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \ge 9
r
(
ab
+
b
c
+
c
a
)
+
(
3
−
r
)
(
a
1
+
b
1
+
c
1
)
≥
9
is true for
a
,
b
,
c
a,b,c
a
,
b
,
c
arbitrary positive reals
5
2
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2p^2q + 45pq^2 a perfect square - Puerto Rico TST 2010.5
Find all prime numbers
p
p
p
and
q
q
q
such that
2
p
2
q
+
45
p
q
2
2p^2q + 45pq^2
2
p
2
q
+
45
p
q
2
is a perfect square.
10 boys and 10 girls in a dance class Puerto Rico Ibero TST 2010.5
In a dance class there are
10
10
10
boys and
10
10
10
girls. It is known that for each
1
≤
k
≤
10
1\le k\le 10
1
≤
k
≤
10
and for each group of
k
k
k
boys, the number of girls who are friends with at least one boy in the group is not less than
k
k
k
. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.
4
2
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min of (3x^2 + 16xy + 15y^2)/(x^2 + y^2) - Puerto Rico TST 2010.4
Find the largest possible value in the real numbers of the term
3
x
2
+
16
x
y
+
15
y
2
x
2
+
y
2
\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}
x
2
+
y
2
3
x
2
+
16
x
y
+
15
y
2
with
x
2
+
y
2
≠
0
x^2 + y^2 \ne 0
x
2
+
y
2
=
0
.
concyclic wanted, AC=AD, orthocenter Puerto Rico Ibero TST 2010.4
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
>
B
C
>
A
C
AB>BC>AC
A
B
>
BC
>
A
C
. Let
D
D
D
be a point different from
C
C
C
on the segment
B
C
BC
BC
, such that
A
C
=
A
D
AC=AD
A
C
=
A
D
. Let
H
H
H
be the orthocenter of triangle
A
B
C
ABC
A
BC
and let
A
1
A_1
A
1
and
B
1
B_1
B
1
be the intersections of the heights from
A
A
A
and
B
B
B
to the opposite sides, respectively. Let
E
E
E
be the intersection of the lines
A
1
B
1
A_1B_1
A
1
B
1
and
D
H
DH
DH
. Prove that
B
B
B
,
D
D
D
,
B
1
B_1
B
1
,
E
E
E
are concyclic.
3
2
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no of different wheels by 5 children - Puerto Rico TST 2010.3
Five children are divided into groups and in each group they take the hand forming a wheel to dance spinning. How many different wheels those children can form, if it is valid that there are groups of
1
1
1
to
5
5
5
children, and can there be any number of groups?
x^2+y^2+1 >= 2(xy-x+y) Puerto Rico Ibero TST 2010.3
Prove that the inequality
x
2
+
y
2
+
1
≥
2
(
x
y
−
x
+
y
)
x^2+y^2+1\ge 2(xy-x+y)
x
2
+
y
2
+
1
≥
2
(
x
y
−
x
+
y
)
is satisfied by any
x
x
x
,
y
y
y
real numbers. Indicate when the equality is satisfied.
2
2
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a_3+a_5=? if 1 a_2 a_3... a_n = n^2 - Puerto Rico TST 2010.2
There is the sequence of numbers
1
,
a
2
,
a
3
,
.
.
.
1, a_2, a_3, ...
1
,
a
2
,
a
3
,
...
such that satisfies
1
⋅
a
2
⋅
a
3
⋅
.
.
.
⋅
a
n
=
n
2
1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2
1
⋅
a
2
⋅
a
3
⋅
...
⋅
a
n
=
n
2
, for every integer
n
>
2
n> 2
n
>
2
. Determine the value of
a
3
+
a
5
a_3 + a_5
a
3
+
a
5
.
two 3-digit numbers wanted Puerto Rico Ibero TST 2010.2
Find two three-digit numbers
x
x
x
and
y
y
y
such that the sum of all other three digit numbers is equal to
600
x
600x
600
x
.