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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2000 Singapore Senior Math Olympiad
2000 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(3)
3
1
Hide problems
sum n_1/[n_i,n_{i+1}] <= 1- 1/2^{1999}, lcm inequality
Let
n
1
,
n
2
,
n
3
,
.
.
.
,
n
2000
n_1,n_2,n_3,...,n_{2000}
n
1
,
n
2
,
n
3
,
...
,
n
2000
be
2000
2000
2000
positive integers satisfying
n
1
<
n
2
<
n
3
<
.
.
.
<
n
2000
n_1<n_2<n_3<...<n_{2000}
n
1
<
n
2
<
n
3
<
...
<
n
2000
. Prove that
n
1
[
n
1
,
n
2
]
+
n
1
[
n
2
,
n
3
]
+
n
1
[
n
3
,
n
4
]
+
.
.
.
+
n
1
[
n
1999
,
n
2000
]
≤
1
−
1
2
1999
\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}
[
n
1
,
n
2
]
n
1
+
[
n
2
,
n
3
]
n
1
+
[
n
3
,
n
4
]
n
1
+
...
+
[
n
1999
,
n
2000
]
n
1
≤
1
−
2
1999
1
where
[
a
,
b
]
[a, b]
[
a
,
b
]
denotes the least common multiple of
a
a
a
and
b
b
b
.
1
1
Hide problems
6 triangles with equal areas
In
△
A
B
C
\vartriangle ABC
△
A
BC
, the points
D
,
E
D, E
D
,
E
and
F
F
F
lie on
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
A
CA
C
A
respectively. The line segments
A
E
,
B
F
AE, BF
A
E
,
BF
and
C
D
CD
C
D
meet at the point
G
G
G
. Suppose that the area of each of
△
B
G
D
,
△
E
C
G
\vartriangle BGD, \vartriangle ECG
△
BG
D
,
△
ECG
and
△
G
F
A
\vartriangle GFA
△
GF
A
is
1
1
1
cm
2
^2
2
. Prove that the area of each of
△
B
E
G
,
△
G
C
F
\vartriangle BEG, \vartriangle GCF
△
BEG
,
△
GCF
and
△
A
D
G
\vartriangle ADG
△
A
D
G
is also
1
1
1
cm
2
^2
2
. https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png
2
1
Hide problems
(m - 1)! + 1 = m^n , m>5 diophantine
Prove that there exist no positive integers
m
m
m
and
n
n
n
such that
m
>
5
m > 5
m
>
5
and
(
m
−
1
)
!
+
1
=
m
n
(m - 1)! + 1 = m^n
(
m
−
1
)!
+
1
=
m
n
.