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Problems
Contests
International Contests
APMO
2004 APMO
2004 APMO
Part of
APMO
Subcontests
(5)
4
1
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Floor function
For a real number
x
x
x
, let
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
stand for the largest integer that is less than or equal to
x
x
x
. Prove that
⌊
(
n
−
1
)
!
n
(
n
+
1
)
⌋
\left\lfloor{(n-1)!\over n(n+1)}\right\rfloor
⌊
n
(
n
+
1
)
(
n
−
1
)!
⌋
is even for every positive integer
n
n
n
.
3
1
Hide problems
Colour the points
Let a set
S
S
S
of 2004 points in the plane be given, no three of which are collinear. Let
L
{\cal L}
L
denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of
S
S
S
with at most two colours, such that for any points
p
,
q
p,q
p
,
q
of
S
S
S
, the number of lines in
L
{\cal L}
L
which separate
p
p
p
from
q
q
q
is odd if and only if
p
p
p
and
q
q
q
have the same colour. Note: A line
ℓ
\ell
ℓ
separates two points
p
p
p
and
q
q
q
if
p
p
p
and
q
q
q
lie on opposite sides of
ℓ
\ell
ℓ
with neither point on
ℓ
\ell
ℓ
.
1
1
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(i+j)/(i,j) belongs to s
Determine all finite nonempty sets
S
S
S
of positive integers satisfying {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, where
(
i
,
j
)
(i,j)
(
i
,
j
)
is the greatest common divisor of
i
i
i
and
j
j
j
.
5
1
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Not homogenous
Prove that the inequality
(
a
2
+
2
)
(
b
2
+
2
)
(
c
2
+
2
)
≥
9
(
a
b
+
b
c
+
c
a
)
\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)
(
a
2
+
2
)
(
b
2
+
2
)
(
c
2
+
2
)
≥
9
(
ab
+
b
c
+
c
a
)
holds for all positive reals
a
a
a
,
b
b
b
,
c
c
c
.
2
1
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Areas of triangles AOH, BOH, COH
Let
O
O
O
be the circumcenter and
H
H
H
the orthocenter of an acute triangle
A
B
C
ABC
A
BC
. Prove that the area of one of the triangles
A
O
H
AOH
A
O
H
,
B
O
H
BOH
BO
H
and
C
O
H
COH
CO
H
is equal to the sum of the areas of the other two.