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International Contests
Rioplatense Mathematical Olympiad, Level 3
2004 Rioplatense Mathematical Olympiad, Level 3
2004 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
3
2
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Geometric inequality with areas of a hexagon and triangle
In a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, triangles
A
C
E
ACE
A
CE
and
B
D
F
BDF
B
D
F
have the same circumradius
R
R
R
. If triangle
A
C
E
ACE
A
CE
has inradius
r
r
r
, prove that
Area
(
A
B
C
D
E
F
)
≤
R
r
⋅
Area
(
A
C
E
)
.
\text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).
Area
(
A
BC
D
EF
)
≤
r
R
⋅
Area
(
A
CE
)
.
Fifth largest numbers in a partition of {1,2,...,900}
Consider a partition of
{
1
,
2
,
…
,
900
}
\{1,2,\ldots,900\}
{
1
,
2
,
…
,
900
}
into
30
30
30
subsets
S
1
,
S
2
,
…
,
S
30
S_1,S_2,\ldots,S_{30}
S
1
,
S
2
,
…
,
S
30
each with
30
30
30
elements. In each
S
k
S_k
S
k
, we paint the fifth largest number blue. Is it possible that, for
k
=
1
,
2
,
…
,
30
k=1,2,\ldots,30
k
=
1
,
2
,
…
,
30
, the sum of the elements of
S
k
S_k
S
k
exceeds the sum of the blue numbers?
2
2
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Subsets of {1,...,2004} with a & b such that 2004 | a^2-b^2
Find the smallest integer
n
n
n
such that each subset of
{
1
,
2
,
…
,
2004
}
\{1,2,\ldots, 2004\}
{
1
,
2
,
…
,
2004
}
with
n
n
n
elements has two distinct elements
a
a
a
and
b
b
b
for which
a
2
−
b
2
a^2-b^2
a
2
−
b
2
is a multiple of
2004
2004
2004
.
Arranging cardboard circles on a table
A collection of cardboard circles, each with a diameter of at most
1
1
1
, lie on a
5
×
8
5\times 8
5
×
8
table without overlapping or overhanging the edge of the table. A cardboard circle of diameter
2
2
2
is added to the collection. Prove that this new collection of cardboard circles can be placed on a
7
×
7
7\times 7
7
×
7
table without overlapping or overhanging the edge.
1
2
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Functional equation: x p(y/x) + y p(x/y) = x + y
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
x
P
(
y
x
)
+
y
P
(
x
y
)
=
x
+
y
xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y
x
P
(
x
y
)
+
y
P
(
y
x
)
=
x
+
y
for all nonzero real numbers
x
x
x
and
y
y
y
.
Integers n such that n divides x^13-x for all x in N
How many integers
n
>
1
n>1
n
>
1
are there such that
n
n
n
divides
x
13
−
x
x^{13}-x
x
13
−
x
for every positive integer
x
x
x
?