MathDB
Problems
Contests
International Contests
CentroAmerican
2016 CentroAmerican
2016 CentroAmerican
Part of
CentroAmerican
Subcontests
(6)
5
1
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Irie numbers
We say a number is irie if it can be written in the form
1
+
1
k
1+\dfrac{1}{k}
1
+
k
1
for some positive integer
k
k
k
. Prove that every integer
n
≥
2
n \geq 2
n
≥
2
can be written as the product of
r
r
r
distinct irie numbers for every integer
r
≥
n
−
1
r \geq n-1
r
≥
n
−
1
.
4
1
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Game with coprime integers
The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is
n
n
n
, the player in his/her turn must replace it by an integer
m
m
m
coprime with
n
n
n
and such that
n
<
m
<
n
2
n<m<n^2
n
<
m
<
n
2
. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.
6
1
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Equal radius
Let
△
A
B
C
\triangle ABC
△
A
BC
be triangle with incenter
I
I
I
and circumcircle
Γ
\Gamma
Γ
. Let
M
=
B
I
∩
Γ
M=BI\cap \Gamma
M
=
B
I
∩
Γ
and
N
=
C
I
∩
Γ
N=CI\cap \Gamma
N
=
C
I
∩
Γ
, the line parallel to
M
N
MN
MN
through
I
I
I
cuts
A
B
AB
A
B
,
A
C
AC
A
C
in
P
P
P
and
Q
Q
Q
. Prove that the circumradius of
⊙
(
B
N
P
)
\odot (BNP)
⊙
(
BNP
)
and
⊙
(
C
M
Q
)
\odot (CMQ)
⊙
(
CMQ
)
are equal.
3
1
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Cyclic permutation of roots
The polynomial
Q
(
x
)
=
x
3
−
21
x
+
35
Q(x)=x^3-21x+35
Q
(
x
)
=
x
3
−
21
x
+
35
has three different real roots. Find real numbers
a
a
a
and
b
b
b
such that the polynomial
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
cyclically permutes the roots of
Q
Q
Q
, that is, if
r
r
r
,
s
s
s
and
t
t
t
are the roots of
Q
Q
Q
(in some order) then
P
(
r
)
=
s
P(r)=s
P
(
r
)
=
s
,
P
(
s
)
=
t
P(s)=t
P
(
s
)
=
t
and
P
(
t
)
=
r
P(t)=r
P
(
t
)
=
r
.
2
1
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Midpoints and foot of altitude are collinear
Let
A
B
C
ABC
A
BC
be an acute-angled triangle,
Γ
\Gamma
Γ
its circumcircle and
M
M
M
the midpoint of
B
C
BC
BC
. Let
N
N
N
be a point in the arc
B
C
BC
BC
of
Γ
\Gamma
Γ
not containing
A
A
A
such that
∠
N
A
C
=
∠
B
A
M
\angle NAC= \angle BAM
∠
N
A
C
=
∠
B
A
M
. Let
R
R
R
be the midpoint of
A
M
AM
A
M
,
S
S
S
the midpoint of
A
N
AN
A
N
and
T
T
T
the foot of the altitude through
A
A
A
. Prove that
R
R
R
,
S
S
S
and
T
T
T
are collinear.
1
1
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Perfect square digits
Find all positive integers
n
n
n
that have 4 digits, all of them perfect squares, and such that
n
n
n
is divisible by 2, 3, 5 and 7.