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International Contests
Rioplatense Mathematical Olympiad, Level 3
2000 Rioplatense Mathematical Olympiad, Level 3
2000 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
4
1
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a,b,c positive integers, a^2 + b^2 + 1 = c^2, prove [a/2] + [c / 2] is even
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive integers such that
a
2
+
b
2
+
1
=
c
2
a^2 + b^2 + 1 = c^2
a
2
+
b
2
+
1
=
c
2
. Prove that
[
a
/
2
]
+
[
c
/
2
]
[a/2] + [c / 2]
[
a
/2
]
+
[
c
/2
]
is even.Note:
[
x
]
[x]
[
x
]
is the integer part of
x
x
x
.
1
1
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b^2+(b +1)^2+...+(b + a)^2 -3 mutiple of 5, a+b = odd, digit of units of (a+b)?
Let
a
a
a
and
b
b
b
be positive integers such that the number
b
2
+
(
b
+
1
)
2
+
.
.
.
+
(
b
+
a
)
2
−
3
b^2 + (b +1)^2 +...+ (b + a)^2-3
b
2
+
(
b
+
1
)
2
+
...
+
(
b
+
a
)
2
−
3
is multiple of
5
5
5
and
a
+
b
a + b
a
+
b
is odd. Calculate the digit of the units of the number
a
+
b
a + b
a
+
b
written in decimal notation.
2
1
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D,E,F on sides BC, CA, AB so that (AFE)=(BFD)=(CDE), prove (DEF)/(ABC)>=1/4
In a triangle
A
B
C
ABC
A
BC
, points
D
,
E
D, E
D
,
E
and
F
F
F
are considered on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
respectively, such that the areas of the triangles
A
F
E
,
B
F
D
AFE, BFD
A
FE
,
BF
D
and
C
D
E
CDE
C
D
E
are equal. Prove that
(
D
E
F
)
(
A
B
C
)
≥
1
4
\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}
(
A
BC
)
(
D
EF
)
≥
4
1
Note:
(
X
Y
Z
)
(XYZ)
(
X
Y
Z
)
is the area of triangle
X
Y
Z
XYZ
X
Y
Z
.
6
1
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No functions
Let
g
(
x
)
=
a
x
2
+
b
x
+
c
g(x) = ax^2 + bx + c
g
(
x
)
=
a
x
2
+
b
x
+
c
a quadratic function with real coefficients such that the equation
g
(
g
(
x
)
)
=
x
g(g(x)) = x
g
(
g
(
x
))
=
x
has four distinct real roots. Prove that there isn't a function
f
f
f
:
R
−
−
R
R--R
R
−
−
R
such that
f
(
f
(
x
)
)
=
g
(
x
)
f(f(x)) = g(x)
f
(
f
(
x
))
=
g
(
x
)
for all
x
x
x
real
5
1
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Concurrency(little hard)
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
, let
L
L
L
be midpoint of arc
B
C
BC
BC
(the point
A
A
A
is not in this arc) of the circumcircle
w
w
w
(
A
B
C
ABC
A
BC
). Let
E
E
E
be a point in
A
C
AC
A
C
where
A
E
=
A
B
+
A
C
2
AE = \frac{AB + AC}{2}
A
E
=
2
A
B
+
A
C
, the line
E
L
EL
E
L
intersects
w
w
w
in
P
P
P
. If
M
M
M
and
N
N
N
are the midpoints of
A
B
AB
A
B
and
B
C
BC
BC
, respectively, prove that
A
L
,
B
P
AL, BP
A
L
,
BP
and
M
N
MN
MN
are concurrents
3
1
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Rioplatense Olympiad 2001 P3
Let
n
>
1
n>1
n
>
1
be an integer. For each numbers
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2,\dots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
with
x
1
2
+
x
2
2
+
x
3
2
+
⋯
+
x
n
2
=
1
x_1^2+x_2^2+x_3^2+\dots +x_n^2=1
x
1
2
+
x
2
2
+
x
3
2
+
⋯
+
x
n
2
=
1
, denote
m
=
min
{
∣
x
i
−
x
j
∣
,
0
<
i
<
j
<
n
+
1
}
m=\min\{|x_i-x_j|, 0<i<j<n+1\}
m
=
min
{
∣
x
i
−
x
j
∣
,
0
<
i
<
j
<
n
+
1
}
Find the maximum value of
m
m
m
.