MathDB
Problems
Contests
International Contests
IberoAmerican
1999 IberoAmerican
1999 IberoAmerican
Part of
IberoAmerican
Subcontests
(3)
2
2
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14th ibmo - cuba 1999/q2.
Given two circle
M
M
M
and
N
N
N
, we say that
M
M
M
bisects
N
N
N
if they intersect in two points and the common chord is a diameter of
N
N
N
. Consider two fixed non-concentric circles
C
1
C_1
C
1
and
C
2
C_2
C
2
. a) Show that there exists infinitely many circles
B
B
B
such that
B
B
B
bisects both
C
1
C_1
C
1
and
C
2
C_2
C
2
. b) Find the locus of the centres of such circles
B
B
B
.
14th ibmo - cuba 1999/q5.
An acute triangle
△
A
B
C
\triangle{ABC}
△
A
BC
is inscribed in a circle with centre
O
O
O
. The altitudes of the triangle are
A
D
,
B
E
AD,BE
A
D
,
BE
and
C
F
CF
CF
. The line
E
F
EF
EF
cut the circumference on
P
P
P
and
Q
Q
Q
. a) Show that
O
A
OA
O
A
is perpendicular to
P
Q
PQ
PQ
. b) If
M
M
M
is the midpoint of
B
C
BC
BC
, show that
A
P
2
=
2
A
D
⋅
O
M
AP^2=2AD\cdot{OM}
A
P
2
=
2
A
D
⋅
OM
.
3
2
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14th ibmo - cuba 1999/q3.
Let
P
1
,
P
2
,
…
,
P
n
P_1,P_2,\dots,P_n
P
1
,
P
2
,
…
,
P
n
be
n
n
n
distinct points over a line in the plane (
n
≥
2
n\geq2
n
≥
2
). Consider all the circumferences with diameters
P
i
P
j
P_iP_j
P
i
P
j
(
1
≤
i
,
j
≤
n
1\leq{i,j}\leq{n}
1
≤
i
,
j
≤
n
) and they are painted with
k
k
k
given colors. Lets call this configuration a (
n
,
k
n,k
n
,
k
)-cloud.For each positive integer
k
k
k
, find all the positive integers
n
n
n
such that every possible (
n
,
k
n,k
n
,
k
)-cloud has two mutually exterior tangent circumferences of the same color.
14th ibmo - cuba 1999/q6.
Let
A
A
A
and
B
B
B
points in the plane and
C
C
C
a point in the perpendiclar bisector of
A
B
AB
A
B
. It is constructed a sequence of points
C
1
,
C
2
,
…
,
C
n
,
…
C_1,C_2,\dots, C_n,\dots
C
1
,
C
2
,
…
,
C
n
,
…
in the following way:
C
1
=
C
C_1=C
C
1
=
C
and for
n
≥
1
n\geq1
n
≥
1
, if
C
n
C_n
C
n
does not belongs to
A
B
AB
A
B
, then
C
n
+
1
C_{n+1}
C
n
+
1
is the circumcentre of the triangle
△
A
B
C
n
\triangle{ABC_n}
△
A
B
C
n
.Find all the points
C
C
C
such that the sequence
C
1
,
C
2
,
…
C_1,C_2,\dots
C
1
,
C
2
,
…
is defined for all
n
n
n
and turns eventually periodic.Note: A sequence
C
1
,
C
2
,
…
C_1,C_2, \dots
C
1
,
C
2
,
…
is called eventually periodic if there exist positive integers
k
k
k
and
p
p
p
such that
C
n
+
p
=
c
n
C_{n+p}=c_n
C
n
+
p
=
c
n
for all
n
≥
k
n\geq{k}
n
≥
k
.
1
2
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14th ibmo - cuba 1999/q4.
Let
B
B
B
be an integer greater than 10 such that everyone of its digits belongs to the set
{
1
,
3
,
7
,
9
}
\{1,3,7,9\}
{
1
,
3
,
7
,
9
}
. Show that
B
B
B
has a prime divisor greater than or equal to 11.
14th ibmo - cuba 1999/q1.
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.