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Problems
Contests
International Contests
IberoAmerican
1991 IberoAmerican
1991 IberoAmerican
Part of
IberoAmerican
Subcontests
(6)
6
1
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Construction given midpoints and orthocenter
Let
M
M
M
,
N
N
N
and
P
P
P
be three non-collinear points. Construct using straight edge and compass a triangle for which
M
M
M
and
N
N
N
are the midpoints of two of its sides, and
P
P
P
is its orthocenter.
5
1
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Values of a polynomial
Let
P
(
x
,
y
)
=
2
x
2
−
6
x
y
+
5
y
2
P(x,\, y)=2x^{2}-6xy+5y^{2}
P
(
x
,
y
)
=
2
x
2
−
6
x
y
+
5
y
2
. Let us say an integer number
a
a
a
is a value of
P
P
P
if there exist integer numbers
b
b
b
,
c
c
c
such that
P
(
b
,
c
)
=
a
P(b,\, c)=a
P
(
b
,
c
)
=
a
. a) Find all values of
P
P
P
lying between 1 and 100. b) Show that if
r
r
r
and
s
s
s
are values of
P
P
P
, then so is
r
s
rs
rs
.
4
1
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Five-digit number
Find a positive integer
n
n
n
with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of
n
n
n
.
2
1
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Square cut by perpendicular lines
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.
1
1
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Numbers in a cube
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
3
1
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Find f(18/1991)
Let
f
:
[
0
,
1
]
→
R
f: \ [0,\ 1] \rightarrow \mathbb{R}
f
:
[
0
,
1
]
→
R
be an increasing function satisfying the following conditions: a)
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
; b)
f
(
x
3
)
=
f
(
x
)
2
f\left(\frac{x}{3}\right)=\frac{f(x)}{2}
f
(
3
x
)
=
2
f
(
x
)
; c)
f
(
1
−
x
)
=
1
−
f
(
x
)
f(1-x)=1-f(x)
f
(
1
−
x
)
=
1
−
f
(
x
)
. Determine
f
(
18
1991
)
f\left(\frac{18}{1991}\right)
f
(
1991
18
)
.