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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2008 Singapore Senior Math Olympiad
2008 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(4)
5
1
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(1+a^2)(1+b^2)(1+c^2)/(1+a)(1+b)(1+c)>= \frac12 (1+abc) for a,b,c>=0
Let
a
,
b
,
c
≥
0
a,b,c \ge 0
a
,
b
,
c
≥
0
. Prove that
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
≥
1
2
(
1
+
a
b
c
)
\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
≥
2
1
(
1
+
ab
c
)
4
1
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11 committees, each has 5 members, every 2 have a member in common
There are
11
11
11
committees in a club. Each committee has
5
5
5
members and every two committees have a member in common. Show that there is a member who belongs to
4
4
4
committees.
1
1
Hide problems
equal angles lead to equal angles inside a trapezium
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium with
A
D
/
/
B
C
AD // BC
A
D
//
BC
. Suppose
K
K
K
and
L
L
L
are, respectively, points on the sides
A
B
AB
A
B
and
C
D
CD
C
D
such that
∠
B
A
L
=
∠
C
D
K
\angle BAL = \angle CDK
∠
B
A
L
=
∠
C
DK
. Prove that
∠
B
L
A
=
∠
C
K
D
\angle BLA = \angle CKD
∠
B
L
A
=
∠
C
KD
.
3
1
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Singapore Senior 2008
Let there's a function
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
Find all functions
f
f
f
that satisfies: a) f(2u)\equal{}f(u\plus{}v)f(v\minus{}u)\plus{}f(u\minus{}v)f(\minus{}u\minus{}v) b)
f
(
u
)
≥
0
f(u)\geq0
f
(
u
)
≥
0