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Contests
National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2007 Argentina Team Selection Test
2007 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(6)
6
1
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Expression with sum of digits (bounded?)
For natural
n
n
n
we define
s
(
n
)
s(n)
s
(
n
)
as the sum of digits of
n
n
n
(in base ten) Does there exist a positive real constant
c
c
c
such that for all natural
n
n
n
we have
s
(
n
)
s
(
n
2
)
≤
c
\frac{s(n)}{s(n^2)} \le c
s
(
n
2
)
s
(
n
)
≤
c
?
5
1
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Find divisors of n
Let
d
1
,
d
2
…
,
d
r
d_1,d_2 \ldots, d_r
d
1
,
d
2
…
,
d
r
be the positive divisors of
n
n
n
1\equal{}d_1
d
17
d_{17}
d
17
4
1
Hide problems
solve real equation
Find all real values of
x
>
1
x>1
x
>
1
which satisfy: \frac{x^2}{x\minus{}1} \plus{} \sqrt{x\minus{}1} \plus{}\frac{\sqrt{x\minus{}1}}{x^2} \equal{} \frac{x\minus{}1}{x^2} \plus{} \frac{1}{\sqrt{x\minus{}1}} \plus{} \frac{x^2}{\sqrt{x\minus{}1}}
2
1
Hide problems
Prove PI=BQ
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium of parallel sides
A
D
AD
A
D
and
B
C
BC
BC
and non-parallel sides
A
B
AB
A
B
and
C
D
CD
C
D
Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. It is known that exists a point
Q
∈
A
D
Q \in AD
Q
∈
A
D
with
Q
≠
A
Q \neq A
Q
=
A
and
Q
≠
D
Q \neq D
Q
=
D
such that if
P
P
P
is a point of the intersection of the bisectors of
C
Q
D
^
\widehat{ CQD}
CQ
D
and
C
A
D
^
\widehat{CAD}
C
A
D
then
P
I
∥
A
D
PI \parallel AD
P
I
∥
A
D
Prove that PI\equal{}BQ
1
1
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Find posible values of gcd
Let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
be distinct positive integers having exactly two digits in such a way that: X\equal{}10a \plus{}b Y\equal{}10b \plus{}c Z\equal{}10c \plus{}a (
a
,
b
,
c
a,b,c
a
,
b
,
c
are digits) Find all posible values of
g
c
d
(
X
,
Y
,
Z
)
gcd(X,Y,Z)
g
c
d
(
X
,
Y
,
Z
)
3
1
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Coloring dominoes in three colors
A
3000
×
3000
3000\times 3000
3000
×
3000
square is tiled by dominoes (i. e.
1
×
2
1\times 2
1
×
2
rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe
d
d
d
has at most two neighbours of the same color as
d
d
d
. (Two dominoes are said to be neighbours if a cell of one domino has a common edge with a cell of the other one.)