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Problems
Contests
International Contests
APMO
1996 APMO
1996 APMO
Part of
APMO
Subcontests
(5)
5
1
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Triangle inequality
Let
a
a
a
,
b
b
b
,
c
c
c
be the lengths of the sides of a triangle. Prove that
a
+
b
−
c
+
b
+
c
−
a
+
c
+
a
−
b
≤
a
+
b
+
c
\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}
a
+
b
−
c
+
b
+
c
−
a
+
c
+
a
−
b
≤
a
+
b
+
c
and determine when equality occurs.
2
1
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Inequality with !
Let
m
m
m
and
n
n
n
be positive integers such that
n
≤
m
n \leq m
n
≤
m
. Prove that
2
n
n
!
≤
(
m
+
n
)
!
(
m
−
n
)
!
≤
(
m
2
+
m
)
n
2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n
2
n
n
!
≤
(
m
−
n
)!
(
m
+
n
)!
≤
(
m
2
+
m
)
n
4
1
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Form groups
The National Marriage Council wishes to invite
n
n
n
couples to form 17 discussion groups under the following conditions: (1) All members of a group must be of the same sex; i.e. they are either all male or all female. (2) The difference in the size of any two groups is 0 or 1. (3) All groups have at least 1 member. (4) Each person must belong to one and only one group. Find all values of
n
n
n
,
n
≤
1996
n \leq 1996
n
≤
1996
, for which this is possible. Justify your answer.
1
1
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Distance constant
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral
A
B
=
B
C
=
C
D
=
D
A
AB = BC = CD = DA
A
B
=
BC
=
C
D
=
D
A
. Let
M
N
MN
MN
and
P
Q
PQ
PQ
be two segments perpendicular to the diagonal
B
D
BD
B
D
and such that the distance between them is
d
>
B
D
2
d > \frac{BD}{2}
d
>
2
B
D
, with
M
∈
A
D
M \in AD
M
∈
A
D
,
N
∈
D
C
N \in DC
N
∈
D
C
,
P
∈
A
B
P \in AB
P
∈
A
B
, and
Q
∈
B
C
Q \in BC
Q
∈
BC
. Show that the perimeter of hexagon
A
M
N
C
Q
P
AMNCQP
A
MNCQP
does not depend on the position of
M
N
MN
MN
and
P
Q
PQ
PQ
so long as the distance between them remains constant.
3
1
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ABCD cyclic --> incenters of BCD, etc. form rectangle
If
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral, then prove that the incenters of the triangles
A
B
C
ABC
A
BC
,
B
C
D
BCD
BC
D
,
C
D
A
CDA
C
D
A
,
D
A
B
DAB
D
A
B
are the vertices of a rectangle.