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International Contests
Rioplatense Mathematical Olympiad, Level 3
1997 Rioplatense Mathematical Olympiad, Level 3
1997 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
5
1
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max of x_1x_2 + x_2x_3 + ... + x_{n-1}x_n if sum x_1=1, x_1>0
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
be non-negative numbers
n
≥
3
n\ge3
n
≥
3
such that
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x_1 + x_2 + ... + x_n = 1
x
1
+
x
2
+
...
+
x
n
=
1
. Determine the maximum possible value of the expression
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
x_1x_2 + x_2x_3 + ... + x_{n-1}x_n
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
.
6
1
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f(f(n))=2n
Let
N
N
N
be the set of positive integers. Determine if there is a function
f
:
N
→
N
f: N\to N
f
:
N
→
N
such that
f
(
f
(
n
)
)
=
2
n
f(f(n))=2n
f
(
f
(
n
))
=
2
n
, for all
n
n
n
belongs to
N
N
N
.
3
1
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no of positive divisors in 2^n-1 is greater than n
Prove that there are infinitely many positive integers
n
n
n
such that the number of positive divisors in
2
n
−
1
2^n-1
2
n
−
1
is greater than
n
n
n
.
1
1
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integer polynomial
Find all positive integers
n
n
n
with the following property: there exists a polynomial
P
n
(
x
)
P_n(x)
P
n
(
x
)
of degree
n
n
n
, with integer coefficients, such that
P
n
(
0
)
=
0
P_n(0)=0
P
n
(
0
)
=
0
and
P
n
(
x
)
=
n
P_n(x)=n
P
n
(
x
)
=
n
for
n
n
n
distinct integer solutions.
4
1
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2 circles tangent internally to a third circle and to one secant of that circle
Circles
c
1
c_1
c
1
and
c
2
c_2
c
2
are tangent internally to circle
c
c
c
at points
A
A
A
and
B
B
B
, respectively, as seen in the figure. The inner tangent common to
c
1
c_1
c
1
and
c
2
c_2
c
2
touches these circles in
P
P
P
and
Q
Q
Q
, respectively. Show that the
A
P
AP
A
P
and
B
Q
BQ
BQ
lines intersect the circle
c
c
c
at diametrically opposite points. https://cdn.artofproblemsolving.com/attachments/0/a/9490a4d7ba2038e490a858b14ba21d07377c5d.gif
2
1
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sphere tangent to plane of rhombus ABCD and to 3 edges of a prism with base ABCD
Consider a prism, not necessarily right, whose base is a rhombus
A
B
C
D
ABCD
A
BC
D
with side
A
B
=
5
AB = 5
A
B
=
5
and diagonal
A
C
=
8
AC = 8
A
C
=
8
. A sphere of radius
r
r
r
is tangent to the plane
A
B
C
D
ABCD
A
BC
D
at
C
C
C
and tangent to the edges
A
A
1
AA_1
A
A
1
,
B
B
1
BB _1
B
B
1
and
D
D
1
DD_ 1
D
D
1
of the prism. Calculate
r
r
r
.