MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2024 Singapore MO Open
2024 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(2)
Q2
1
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Classic inequality on distinct positive integers
Let
n
n
n
be a fixed positive integer. Find the minimum value of
x
1
3
+
⋯
+
x
n
3
x
1
+
⋯
+
x
n
\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}
x
1
+
⋯
+
x
n
x
1
3
+
⋯
+
x
n
3
where
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
are distinct positive integers.
Q1
1
Hide problems
Easy right-angled triangle problem
In triangle
A
B
C
ABC
A
BC
,
∠
B
=
9
0
∘
\angle B=90^\circ
∠
B
=
9
0
∘
,
A
B
>
B
C
AB>BC
A
B
>
BC
, and
P
P
P
is the point such that
B
P
=
B
C
BP=BC
BP
=
BC
and
∠
A
P
B
=
9
0
∘
\angle APB=90^\circ
∠
A
PB
=
9
0
∘
, where
P
P
P
and
C
C
C
lie on the same side of
A
B
AB
A
B
. Let
Q
Q
Q
be the point on
A
B
AB
A
B
such that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
, and let
M
M
M
be the midpoint of
Q
C
QC
QC
. Prove that the line through
M
M
M
parallel to
A
P
AP
A
P
passes through the midpoint of
A
B
AB
A
B
.