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Problems
Contests
International Contests
IberoAmerican
1992 IberoAmerican
1992 IberoAmerican
Part of
IberoAmerican
Subcontests
(3)
3
2
Hide problems
An equilateral triangle
Let
A
B
C
ABC
A
BC
be an equilateral triangle of sidelength 2 and let
ω
\omega
ω
be its incircle. a) Show that for every point
P
P
P
on
ω
\omega
ω
the sum of the squares of its distances to
A
A
A
,
B
B
B
,
C
C
C
is 5. b) Show that for every point
P
P
P
on
ω
\omega
ω
it is possible to construct a triangle of sidelengths
A
P
AP
A
P
,
B
P
BP
BP
,
C
P
CP
CP
. Also, the area of such triangle is
3
4
\frac{\sqrt{3}}{4}
4
3
.
Nice geometric inequality
In a triangle
A
B
C
ABC
A
BC
, points
A
1
A_{1}
A
1
and
A
2
A_{2}
A
2
are chosen in the prolongations beyond
A
A
A
of segments
A
B
AB
A
B
and
A
C
AC
A
C
, such that
A
A
1
=
A
A
2
=
B
C
AA_{1}=AA_{2}=BC
A
A
1
=
A
A
2
=
BC
. Define analogously points
B
1
B_{1}
B
1
,
B
2
B_{2}
B
2
,
C
1
C_{1}
C
1
,
C
2
C_{2}
C
2
. If
[
A
B
C
]
[ABC]
[
A
BC
]
denotes the area of triangle
A
B
C
ABC
A
BC
, show that
[
A
1
A
2
B
1
B
2
C
1
C
2
]
≥
13
[
A
B
C
]
[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]
[
A
1
A
2
B
1
B
2
C
1
C
2
]
≥
13
[
A
BC
]
.
2
2
Hide problems
f(x)>1
Given the positive real numbers
a
1
<
a
2
<
⋯
<
a
n
a_{1}<a_{2}<\cdots<a_{n}
a
1
<
a
2
<
⋯
<
a
n
, consider the function
f
(
x
)
=
a
1
x
+
a
1
+
a
2
x
+
a
2
+
⋯
+
a
n
x
+
a
n
f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}
f
(
x
)
=
x
+
a
1
a
1
+
x
+
a
2
a
2
+
⋯
+
x
+
a
n
a
n
Determine the sum of the lengths of the disjoint intervals formed by all the values of
x
x
x
such that
f
(
x
)
>
1
f(x)>1
f
(
x
)
>
1
.
Construction of a trapezoid
Given a circle
Γ
\Gamma
Γ
and the positive numbers
h
h
h
and
m
m
m
, construct with straight edge and compass a trapezoid inscribed in
Γ
\Gamma
Γ
, such that it has altitude
h
h
h
and the sum of its parallel sides is
m
m
m
.
1
2
Hide problems
Last digit of 1+2+...+n
For every positive integer
n
n
n
we define
a
n
a_{n}
a
n
as the last digit of the sum
1
+
2
+
⋯
+
n
1+2+\cdots+n
1
+
2
+
⋯
+
n
. Compute
a
1
+
a
2
+
⋯
+
a
1992
a_{1}+a_{2}+\cdots+a_{1992}
a
1
+
a
2
+
⋯
+
a
1992
.
Two sequences
Let
{
a
n
}
n
≥
0
\{a_{n}\}_{n \geq 0}
{
a
n
}
n
≥
0
and
{
b
n
}
n
≥
0
\{b_{n}\}_{n \geq 0}
{
b
n
}
n
≥
0
be two sequences of integer numbers such that: i.
a
0
=
0
a_{0}=0
a
0
=
0
,
b
0
=
8
b_{0}=8
b
0
=
8
. ii. For every
n
≥
0
n \geq 0
n
≥
0
,
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
a_{n+2}=2a_{n+1}-a_{n}+2
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
,
b
n
+
2
=
2
b
n
+
1
−
b
n
b_{n+2}=2b_{n+1}-b_{n}
b
n
+
2
=
2
b
n
+
1
−
b
n
. iii.
a
n
2
+
b
n
2
a_{n}^{2}+b_{n}^{2}
a
n
2
+
b
n
2
is a perfect square for every
n
≥
0
n \geq 0
n
≥
0
. Find at least two values of the pair
(
a
1992
,
b
1992
)
(a_{1992},\, b_{1992})
(
a
1992
,
b
1992
)
.