MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2017 Singapore MO Open
2017 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(5)
2
1
Hide problems
sum (a_i-b_i) (a_i (sum a_1^2)^{p/2}-b_i (sum b_i^2)^{p/2} >=0
Let
a
1
,
a
2
,
.
.
.
,
a
n
,
b
1
,
b
2
,
.
.
.
,
b
n
,
p
a_1,a_2,...,a_n,b_1,b_2,...,b_n,p
a
1
,
a
2
,
...
,
a
n
,
b
1
,
b
2
,
...
,
b
n
,
p
be real numbers with
p
>
−
1
p >- 1
p
>
−
1
. Prove that
∑
i
=
1
n
(
a
i
−
b
i
)
(
a
i
(
a
1
2
+
a
2
2
+
.
.
.
+
a
n
2
)
p
/
2
−
b
i
(
b
1
2
+
b
2
2
+
.
.
.
+
b
n
2
)
p
/
2
)
≥
0
\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0
i
=
1
∑
n
(
a
i
−
b
i
)
(
a
i
(
a
1
2
+
a
2
2
+
...
+
a
n
2
)
p
/2
−
b
i
(
b
1
2
+
b
2
2
+
...
+
b
n
2
)
p
/2
)
≥
0
5
1
Hide problems
1 to n^2 label 2 nxn square arrays, red color available
Let
A
A
A
and
B
B
B
be two
n
×
n
n \times n
n
×
n
square arrays. The cells of
A
A
A
are labelled by the numbers from
1
1
1
to
n
2
n^2
n
2
from left to right starting from the top row, whereas the cells of
B
B
B
are labelled by the numbers from
1
1
1
to
n
2
n^2
n
2
along rising north-easterly diagonals starting with the upper left-hand corner. Stack the array
B
B
B
on top of the array
A
A
A
. If two overlapping cells have the same number, they are coloured red. Determine those
n
n
n
for which there is at least one red cell other than the cells at top left corner, bottom right corner and the centre (when
n
n
n
is odd). Below shows the arrays for
n
=
4
n=4
n
=
4
. https://cdn.artofproblemsolving.com/attachments/8/e/cc8a435cb28420ccf91340023d440e39f0e849.png
4
1
Hide problems
x_1<y_1<x_2<y_2<...<x_n<y_n, where x_i geom. and y_i arithm. progression
Let
n
>
3
n > 3
n
>
3
be an integer. Prove that there exist positive integers
x
1
,
.
.
.
,
x
n
x_1,..., x_n
x
1
,
...
,
x
n
in geometric progression and positive integers
y
1
,
.
.
.
,
y
n
y_1,..., y_n
y
1
,
...
,
y
n
in arithmetic progression such that
x
1
<
y
1
<
x
2
<
y
2
<
.
.
.
<
x
n
<
y
n
x_1<y_1<x_2<y_2<...<x_n<y_n
x
1
<
y
1
<
x
2
<
y
2
<
...
<
x
n
<
y
n
3
1
Hide problems
\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.
Find the smallest positive integer
n
n
n
so that
1
2
+
2
2
+
.
.
.
+
n
2
n
\sqrt{\frac{1^2+2^2+...+n^2}{n}}
n
1
2
+
2
2
+
...
+
n
2
is an integer.
1
1
Hide problems
midpoint lies on circumcircle, incircle and a third circle related
The incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
touches the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. A circle through
A
A
A
and
B
B
B
encloses
△
A
B
C
\vartriangle ABC
△
A
BC
and intersects the line
D
E
DE
D
E
at points
P
P
P
and
Q
Q
Q
. Prove that the midpoint of
A
B
AB
A
B
lies on the circumircle of
△
P
Q
F
\vartriangle PQF
△
PQF
.