MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2001 Singapore MO Open
2001 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(4)
2
1
Hide problems
sum a_k^4/(a_k^2+a_{k+1}^2}) >= 1/2n when sum a_i =1, a_i>0
Let
n
n
n
be a positive integer, and let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be
n
n
n
positive real numbers such that
a
1
+
a
2
+
.
.
.
+
a
n
=
1
a_1+a_2+...+a_n = 1
a
1
+
a
2
+
...
+
a
n
=
1
. Is it true that
a
1
4
a
1
2
+
a
2
2
+
a
2
4
a
2
2
+
a
3
2
+
a
3
4
a
3
2
+
a
4
2
+
.
.
.
+
a
n
−
1
4
a
n
−
1
2
+
a
n
2
+
a
n
4
a
n
2
+
a
1
2
≥
1
2
n
\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}
a
1
2
+
a
2
2
a
1
4
+
a
2
2
+
a
3
2
a
2
4
+
a
3
2
+
a
4
2
a
3
4
+
...
+
a
n
−
1
2
+
a
n
2
a
n
−
1
4
+
a
n
2
+
a
1
2
a
n
4
≥
2
n
1
? Justify your answer.
4
1
Hide problems
N^2 as sum of the squares of n consecutive positive intege
A positive integer
n
n
n
is said to possess Property (
A
A
A
) if there exists a positive integer
N
N
N
such that
N
2
N^2
N
2
can be written as the sum of the squares of
n
n
n
consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property (
A
A
A
)? Justify your answer. (As an example, the number
n
=
2
n = 2
n
=
2
possesses Property (
A
A
A
) since
5
2
=
3
2
+
4
2
5^2 = 3^2 + 4^2
5
2
=
3
2
+
4
2
).
3
1
Hide problems
2001 golf balls which are numbered from 1 to 2001, a few in a box
Suppose that there are
2001
2001
2001
golf balls which are numbered from
1
1
1
to
2001
2001
2001
respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither
5
5
5
nor
8
8
8
. How many such golf balls the box can contain at most? Justify your answer.
1
1
Hide problems
computational inside a parallelogram, orthocenter related
In a parallelogram
A
B
C
D
ABCD
A
BC
D
, the perpendiculars from
A
A
A
to
B
C
BC
BC
and
C
D
CD
C
D
meet the line segments
B
C
BC
BC
and
C
D
CD
C
D
at the points
E
E
E
and
F
F
F
respectively. Suppose
A
C
=
37
AC = 37
A
C
=
37
cm and
E
F
=
35
EF = 35
EF
=
35
cm. Let
H
H
H
be the orthocentre of
△
A
E
F
\vartriangle AEF
△
A
EF
. Find the length of
A
H
AH
A
H
in cm. Show the steps in your calculations.