MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2012 Singapore Senior Math Olympiad
2012 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(5)
4
1
Hide problems
|a_k| \leq \frac{k(n+1-k)}{2} when |a_{k-1} - 2a_{k} + a_{k+1}| \leq 1
Let
a
1
,
a
2
,
.
.
.
,
a
n
,
a
n
+
1
a_1, a_2, ..., a_n, a_{n+1}
a
1
,
a
2
,
...
,
a
n
,
a
n
+
1
be a finite sequence of real numbers satisfying
a
0
=
a
n
+
1
=
0
a_0 = a_{n+1} = 0
a
0
=
a
n
+
1
=
0
and
∣
a
k
−
1
−
2
a
k
+
a
k
+
1
∣
≤
1
|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1
∣
a
k
−
1
−
2
a
k
+
a
k
+
1
∣
≤
1
for
k
=
1
,
2
,
.
.
.
,
n
k = 1, 2, ..., n
k
=
1
,
2
,
...
,
n
Prove that for
k
=
0
,
1
,
.
.
.
,
n
+
1
,
k=0, 1, ..., n+1,
k
=
0
,
1
,
...
,
n
+
1
,
∣
a
k
∣
≤
k
(
n
+
1
−
k
)
2
|a_k| \leq \frac{k(n+1-k)}{2}
∣
a
k
∣
≤
2
k
(
n
+
1
−
k
)
3
1
Hide problems
46 squares are colored red in a 9x9 board
If
46
46
46
squares are colored red in a
9
×
9
9\times 9
9
×
9
board, show that there is a
2
×
2
2\times 2
2
×
2
block on the board in which at least
3
3
3
of the squares are colored red.
2
1
Hide problems
n equals the square of the sum of the digits of n
Determine all positive integers
n
n
n
such that
n
n
n
equals the square of the sum of the digits of
n
n
n
.
5
1
Hide problems
(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) <= 25 when a+b=c+d=2, a,b,c,d >=0
For
a
,
b
,
c
,
d
≥
0
a,b,c,d \geq 0
a
,
b
,
c
,
d
≥
0
with
a
+
b
=
c
+
d
=
2
a + b = c + d = 2
a
+
b
=
c
+
d
=
2
, prove
(
a
2
+
c
2
)
(
a
2
+
d
2
)
(
b
2
+
c
2
)
(
b
2
+
d
2
)
≤
25
(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25
(
a
2
+
c
2
)
(
a
2
+
d
2
)
(
b
2
+
c
2
)
(
b
2
+
d
2
)
≤
25
1
1
Hide problems
pure geometry-prove
A circle
ω
\omega
ω
through the incentre
I
I
I
of a triangle
A
B
C
ABC
A
BC
and tangent to
A
B
AB
A
B
at
A
A
A
, intersects the segment
B
C
BC
BC
at
D
D
D
and the extension of
B
C
BC
BC
at
E
E
E
. Prove that the line
I
C
IC
I
C
intersects
ω
\omega
ω
at a point
M
M
M
such that
M
D
=
M
E
MD=ME
M
D
=
ME
.