MathDB
Problems
Contests
International Contests
APMO
1998 APMO
1998 APMO
Part of
APMO
Subcontests
(5)
5
1
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Find largest n
Find the largest integer
n
n
n
such that
n
n
n
is divisible by all positive integers less than
n
3
\sqrt[3]{n}
3
n
.
2
1
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Can't be power of 2
Show that for any positive integers
a
a
a
and
b
b
b
,
(
36
a
+
b
)
(
a
+
36
b
)
(36a+b)(a+36b)
(
36
a
+
b
)
(
a
+
36
b
)
cannot be a power of
2
2
2
.
3
1
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Product of (1+a/b)
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers. Prove that
(
1
+
a
b
)
(
1
+
b
c
)
(
1
+
c
a
)
≥
2
(
1
+
a
+
b
+
c
a
b
c
3
)
.
\biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr).
(
1
+
b
a
)
(
1
+
c
b
)
(
1
+
a
c
)
≥
2
(
1
+
3
ab
c
a
+
b
+
c
)
.
4
1
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Perpendicular
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
the foot of the altitude from
A
A
A
. Let
E
E
E
and
F
F
F
lie on a line passing through
D
D
D
such that
A
E
AE
A
E
is perpendicular to
B
E
BE
BE
,
A
F
AF
A
F
is perpendicular to
C
F
CF
CF
, and
E
E
E
and
F
F
F
are different from
D
D
D
. Let
M
M
M
and
N
N
N
be the midpoints of the segments
B
C
BC
BC
and
E
F
EF
EF
, respectively. Prove that
A
N
AN
A
N
is perpendicular to
N
M
NM
NM
.
1
1
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Elements of set
Let
F
F
F
be the set of all
n
n
n
-tuples
(
A
1
,
…
,
A
n
)
(A_1, \ldots, A_n)
(
A
1
,
…
,
A
n
)
such that each
A
i
A_{i}
A
i
is a subset of
{
1
,
2
,
…
,
1998
}
\{1, 2, \ldots, 1998\}
{
1
,
2
,
…
,
1998
}
. Let
∣
A
∣
|A|
∣
A
∣
denote the number of elements of the set
A
A
A
. Find
∑
(
A
1
,
…
,
A
n
)
∈
F
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣
\sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n|
(
A
1
,
…
,
A
n
)
∈
F
∑
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣