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International Contests
IberoAmerican
1985 IberoAmerican
1985 IberoAmerican
Part of
IberoAmerican
Subcontests
(3)
3
2
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1st ibmo - colombia 1985./q3
Find all the roots
r
1
r_{1}
r
1
,
r
2
r_{2}
r
2
,
r
3
r_{3}
r
3
y
r
4
r_{4}
r
4
of the equation 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0, knowing that they are real, positive and that: \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.
1st ibmo - colombia 1985./q6
Given an acute triangle
A
B
C
ABC
A
BC
, let
D
D
D
,
E
E
E
and
F
F
F
be points in the lines
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
respectively. If the lines
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
pass through
O
O
O
the centre of the circumcircle of the triangle
A
B
C
ABC
A
BC
, whose radius is
R
R
R
, show that: \frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}
2
2
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1st ibmo - colombia 1985./q2
Let
P
P
P
be a point in the interior of the equilateral triangle
△
A
B
C
\triangle{}ABC
△
A
BC
such that PA \equal{} 5, PB \equal{} 7, PC \equal{} 8. Find the length of the side of the triangle
A
B
C
ABC
A
BC
.
1st ibmo - colombia 1985./q5
To each positive integer
n
n
n
it is assigned a non-negative integer
f
(
n
)
f(n)
f
(
n
)
such that the following conditions are satisfied:(1) f(rs) \equal{} f(r)\plus{}f(s) (2) f(n) \equal{} 0, if the first digit (from right to left) of
n
n
n
is 3. (3) f(10) \equal{} 0.Find
f
(
1985
)
f(1985)
f
(
1985
)
. Justify your answer.
1
2
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1st ibmo - colombia 1985./q1
Find all the triples of integers
(
a
,
b
,
c
)
(a, b,c)
(
a
,
b
,
c
)
such that: \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}
1st ibmo - colombia 1985./q4
If
x
≠
1
x\neq1
x
=
1
,
y
≠
1
y\neq1
y
=
1
,
x
≠
y
x\neq y
x
=
y
and \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y} show that both fractions are equal to x\plus{}y\plus{}z.