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Problems
Contests
International Contests
IberoAmerican
1987 IberoAmerican
1987 IberoAmerican
Part of
IberoAmerican
Subcontests
(3)
3
2
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m must be a perfect square
Prove that if
m
,
n
,
r
m,n,r
m
,
n
,
r
are positive integers, and:
1
+
m
+
n
3
=
(
2
+
3
)
2
r
−
1
1+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1}
1
+
m
+
n
3
=
(
2
+
3
)
2
r
−
1
then
m
m
m
is a perfect square.
PQ forms equal angles with AB and CD
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and let
P
P
P
and
Q
Q
Q
be the points on the sides
A
D
AD
A
D
and
B
C
BC
BC
respectively such that
A
P
P
D
=
B
Q
Q
C
=
A
B
C
D
\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}
P
D
A
P
=
QC
BQ
=
C
D
A
B
. Prove that the line
P
Q
PQ
PQ
forms equal angles with the lines
A
B
AB
A
B
and
C
D
CD
C
D
.
2
2
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AMPN is inscriptable implies ABC is isosceles
In a triangle
A
B
C
ABC
A
BC
,
M
M
M
and
N
N
N
are the respective midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
, and
P
P
P
is the point of intersection of
B
M
BM
BM
and
C
N
CN
CN
. Prove that, if it is possible to inscribe a circle in the quadrilateral
A
M
P
N
AMPN
A
MPN
, then the triangle
A
B
C
ABC
A
BC
is isosceles.
Find arctan r + arctan s +arctan t
Let
r
,
s
,
t
r,s,t
r
,
s
,
t
be the roots of the equation
x
(
x
−
2
)
(
3
x
−
7
)
=
2
x(x-2)(3x-7)=2
x
(
x
−
2
)
(
3
x
−
7
)
=
2
. Show that
r
,
s
,
t
r,s,t
r
,
s
,
t
are real and positive and determine
arctan
r
+
arctan
s
+
arctan
t
\arctan r+\arctan s +\arctan t
arctan
r
+
arctan
s
+
arctan
t
.
1
2
Hide problems
Find the function [IberoAmerican 1987]
Find the function
f
(
x
)
f(x)
f
(
x
)
such that
f
(
x
)
2
f
(
1
−
x
x
+
1
)
=
64
x
f(x)^2f\left(\frac{1-x}{x+1}\right) =64x
f
(
x
)
2
f
(
x
+
1
1
−
x
)
=
64
x
for
x
≠
0
,
x
≠
1
,
x
≠
−
1
x\not=0,x\not=1,x\not=-1
x
=
0
,
x
=
1
,
x
=
−
1
.
Iberoamerican 1987.5
The sequence
(
p
n
)
(p_n)
(
p
n
)
is defined as follows:
p
1
=
2
p_1=2
p
1
=
2
and for all
n
n
n
greater than or equal to
2
2
2
,
p
n
p_n
p
n
is the largest prime divisor of the expression
p
1
p
2
p
3
…
p
n
−
1
+
1
p_1p_2p_3\ldots p_{n-1}+1
p
1
p
2
p
3
…
p
n
−
1
+
1
. Prove that every
p
n
p_n
p
n
is different from
5
5
5
.