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Rioplatense Mathematical Olympiad, Level 3
2002 Rioplatense Mathematical Olympiad, Level 3
2002 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
2
1
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if 0 <\sqrt {2002} - \frac {a}{b} <\frac {\lambda}{ab}, prove \lambda \geq 5
Let
λ
\lambda
λ
be a real number such that the inequality
0
<
2002
−
a
b
<
λ
a
b
0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}
0
<
2002
−
b
a
<
ab
λ
holds for an infinite number of pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers. Prove that
λ
≥
5
\lambda \geq 5
λ
≥
5
.
1
1
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\frac{a^2b+b}{ab^2+9} is an integer
Determine all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers for which
a
2
b
+
b
a
b
2
+
9
\frac{a^2b+b}{ab^2+9}
a
b
2
+
9
a
2
b
+
b
is an integer number.
6
1
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game with n circles and k points, one in each circle, circles deleted, min k
Daniel chooses a positive integer
n
n
n
and tells Ana. With this information, Ana chooses a positive integer
k
k
k
and tells Daniel. Daniel draws
n
n
n
circles on a piece of paper and chooses
k
k
k
different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the
k
k
k
points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of
k
k
k
that allows Ana to achieve her goal regardless of how Daniel chose the
n
n
n
circumferences and the
k
k
k
points.
4
1
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(a+b)/c^2+(c+a)/b^2+(b+c)/a^2 >= 9/(a+b+c) +1/a + 1/b + 1/c
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers. Show that
a
+
b
c
2
+
c
+
a
b
2
+
b
+
c
a
2
≥
9
a
+
b
+
c
+
1
a
+
1
b
+
1
c
\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}
c
2
a
+
b
+
b
2
c
+
a
+
a
2
b
+
c
≥
a
+
b
+
c
9
+
a
1
+
b
1
+
c
1
3
1
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prove that CPQ is equilateral if ABC triangle has <C=60^o
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
6
0
o
\angle C=60^o
∠
C
=
6
0
o
. The point
P
P
P
is the symmetric of
A
A
A
with respect to the point of tangency of the circle inscribed with the side
B
C
BC
BC
. Show that if the perpendicular bisector of the
C
P
CP
CP
segment intersects the line containing the angle - bisector of
∠
B
\angle B
∠
B
at the point
Q
Q
Q
, then the triangle
C
P
Q
CPQ
CPQ
is equilateral.
5
1
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problem 3
A
B
C
ABC
A
BC
is any triangle. Tangent at
C
C
C
to circumcircle (
O
O
O
) of
A
B
C
ABC
A
BC
meets
A
B
AB
A
B
at
M
M
M
. Line perpendicular to
O
M
OM
OM
at
M
M
M
intersects
B
C
BC
BC
at
P
P
P
and
A
C
AC
A
C
at
Q
Q
Q
. P.T.
M
P
=
M
Q
MP=MQ
MP
=
MQ
.