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Contests
International Contests
APMO
1990 APMO
1990 APMO
Part of
APMO
Subcontests
(5)
2
1
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Sum of product
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
be positive real numbers, and let
S
k
S_k
S
k
be the sum of the products of
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
taken
k
k
k
at a time. Show that
S
k
S
n
−
k
≥
(
n
k
)
2
a
1
a
2
⋯
a
n
S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n
S
k
S
n
−
k
≥
(
k
n
)
2
a
1
a
2
⋯
a
n
for
k
=
1
k = 1
k
=
1
,
2
2
2
,
⋯
\cdots
⋯
,
n
−
1
n - 1
n
−
1
.
5
1
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Dissect a haxagon into congruent triangles
Show that for every integer
n
≥
6
n \geq 6
n
≥
6
, there exists a convex hexagon which can be dissected into exactly
n
n
n
congruent triangles.
3
1
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Max of product altitudes
Consider all the triangles
A
B
C
ABC
A
BC
which have a fixed base
A
B
AB
A
B
and whose altitude from
C
C
C
is a constant
h
h
h
. For which of these triangles is the product of its altitudes a maximum?
1
1
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Cyclic quadrilateral
Given triangle
A
B
C
ABC
A
BC
, let
D
D
D
,
E
E
E
,
F
F
F
be the midpoints of
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
respectively and let
G
G
G
be the centroid of the triangle. For each value of
∠
B
A
C
\angle BAC
∠
B
A
C
, how many non-similar triangles are there in which
A
E
G
F
AEGF
A
EGF
is a cyclic quadrilateral?
4
1
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1990 persons
A set of 1990 persons is divided into non-intersecting subsets in such a way that 1. No one in a subset knows all the others in the subset, 2. Among any three persons in a subset, there are always at least two who do not know each other, and 3. For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them. (a) Prove that within each subset, every person has the same number of acquaintances. (b) Determine the maximum possible number of subsets. Note: It is understood that if a person
A
A
A
knows person
B
B
B
, then person
B
B
B
will know person
A
A
A
; an acquaintance is someone who is known. Every person is assumed to know one's self.