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Problems
Contests
International Contests
IberoAmerican
2015 İberoAmerican
2015 İberoAmerican
Part of
IberoAmerican
Subcontests
(6)
6
1
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30 integers on a chalkboard
Beto plays the following game with his computer: initially the computer randomly picks
30
30
30
integers from
1
1
1
to
2015
2015
2015
, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer
k
k
k
and some if the numbers written on the chalkboard, and subtracts
k
k
k
from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all
30
30
30
numbers to
0
0
0
, in which case the game ends. Find the minimal number
n
n
n
such that, regardless of which numbers the computer chooses, Beto can end the game in at most
n
n
n
turns.
5
1
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Quartic equation in 2 variables
Find all pairs of integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
(
b
2
+
7
(
a
−
b
)
)
2
=
a
3
b
(b^2+7(a-b))^2=a^{3}b
(
b
2
+
7
(
a
−
b
)
)
2
=
a
3
b
.
4
1
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Point of concurrence on the altitude
Let
A
B
C
ABC
A
BC
be an acute triangle and let
D
D
D
be the foot of the perpendicular from
A
A
A
to side
B
C
BC
BC
. Let
P
P
P
be a point on segment
A
D
AD
A
D
. Lines
B
P
BP
BP
and
C
P
CP
CP
intersect sides
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
, respectively. Let
J
J
J
and
K
K
K
be the feet of the peroendiculars from
E
E
E
and
F
F
F
to
A
D
AD
A
D
, respectively. Show that
F
K
K
D
=
E
J
J
D
\frac{FK}{KD}=\frac{EJ}{JD}
KD
F
K
=
J
D
E
J
.
3
1
Hide problems
Polynomial roots and rational numbers squared
Let
α
\alpha
α
and
β
\beta
β
be the roots of
x
2
−
q
x
+
1
x^{2} - qx + 1
x
2
−
q
x
+
1
, where
q
q
q
is a rational number larger than
2
2
2
. Let
s
1
=
α
+
β
s_1 = \alpha + \beta
s
1
=
α
+
β
,
t
1
=
1
t_1 = 1
t
1
=
1
, and for all integers
n
≥
2
n \geq 2
n
≥
2
:
s
n
=
α
n
+
β
n
s_n = \alpha^n + \beta^n
s
n
=
α
n
+
β
n
t
n
=
s
n
−
1
+
2
s
n
−
2
+
⋅
⋅
⋅
+
(
n
−
1
)
s
1
+
n
t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n
t
n
=
s
n
−
1
+
2
s
n
−
2
+
⋅
⋅
⋅
+
(
n
−
1
)
s
1
+
n
Prove that, for all odd integers
n
n
n
,
t
n
t_n
t
n
is the square of a rational number.
2
1
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An angle bisector cuts a line at a point with a property
A line
r
r
r
contains the points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
in that order. Let
P
P
P
be a point not in
r
r
r
such that
∠
A
P
B
=
∠
C
P
D
\angle{APB} = \angle{CPD}
∠
A
PB
=
∠
CP
D
. Prove that the angle bisector of
∠
A
P
D
\angle{APD}
∠
A
P
D
intersects the line
r
r
r
at a point
G
G
G
such that:
1
G
A
+
1
G
C
=
1
G
B
+
1
G
D
\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}
G
A
1
+
GC
1
=
GB
1
+
G
D
1
1
1
Hide problems
125 written as a sum of pairwise coprime numbers
The number
125
125
125
can be written as a sum of some pairwise coprime integers larger than
1
1
1
. Determine the largest number of terms that the sum may have.