MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2012 Singapore MO Open
2012 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(5)
5
1
Hide problems
Colouring problem ( NOT COMB.)
There are
2012
2012
2012
distinct points in the plane, each of which is to be coloured using one of
n
n
n
colours, so that the numbers of points of each colour are distinct. A set of
n
n
n
points is said to be multi-coloured if their colours are distinct. Determine
n
n
n
that maximizes the number of multi-coloured sets.
4
1
Hide problems
congruent to mod p
Let
p
p
p
be an odd prime. Prove that
1
p
−
2
+
2
p
−
2
+
⋯
+
(
p
−
1
2
)
p
−
2
≡
2
−
2
p
p
(
m
o
d
p
)
.
1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.
1
p
−
2
+
2
p
−
2
+
⋯
+
(
2
p
−
1
)
p
−
2
≡
p
2
−
2
p
(
mod
p
)
.
3
1
Hide problems
Parity qn.
For each
i
=
1
,
2
,
.
.
N
i=1,2,..N
i
=
1
,
2
,
..
N
, let
a
i
,
b
i
,
c
i
a_i,b_i,c_i
a
i
,
b
i
,
c
i
be integers such that at least one of them is odd. Show that one can find integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
a
i
+
y
b
i
+
z
c
i
xa_i+yb_i+zc_i
x
a
i
+
y
b
i
+
z
c
i
is odd for at least
4
7
N
\frac{4}{7}N
7
4
N
different values of
i
i
i
.
2
1
Hide problems
Functional Equation
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
so that
(
x
+
y
)
(
f
(
x
)
−
f
(
y
)
)
=
(
x
−
y
)
f
(
x
+
y
)
(x+y)(f(x)-f(y))=(x-y)f(x+y)
(
x
+
y
)
(
f
(
x
)
−
f
(
y
))
=
(
x
−
y
)
f
(
x
+
y
)
for all
x
,
y
x,y
x
,
y
that belongs to
R
\mathbb{R}
R
.
1
1
Hide problems
Geometry( SMO 2012)
The incircle with centre
I
I
I
of the triangle
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. The line
I
D
ID
I
D
intersects the segment
E
F
EF
EF
at
K
K
K
. Proof that
A
,
K
,
M
A, K, M
A
,
K
,
M
collinear, where
M
M
M
is the midpoint of
B
C
BC
BC
.