MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2005 Singapore Senior Math Olympiad
2005 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(4)
3
1
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S has two 2-element subsets {x,y} and {u,v} such that x+y=u+v
Let
S
S
S
be a subset of
{
1
,
2
,
3
,
.
.
.
,
24
}
\{1,2,3,...,24\}
{
1
,
2
,
3
,
...
,
24
}
with
n
(
S
)
=
10
n(S)=10
n
(
S
)
=
10
. Show that
S
S
S
has two
2
2
2
-element subsets
{
x
,
y
}
\{x,y\}
{
x
,
y
}
and
{
u
,
v
}
\{u,v\}
{
u
,
v
}
such that
x
+
y
=
u
+
v
x+y=u+v
x
+
y
=
u
+
v
1
1
Hide problems
differences of 2 3-digits numbers is a perfect square
The digits of a
3
3
3
-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a
2
2
2
-digit number and is a perfect square. Find the difference.
2
1
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equal segments lead to equal segments
Consider the nonconvex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
∠
C
>
180
\angle C>180
∠
C
>
180
degrees. Let the side
D
C
DC
D
C
extended to meet
A
B
AB
A
B
at
F
F
F
and the side
B
C
BC
BC
extended to meet
A
D
AD
A
D
at
E
E
E
. A line intersects the interiors of the sides
A
B
,
A
D
,
B
C
,
C
D
AB,AD,BC,CD
A
B
,
A
D
,
BC
,
C
D
at points
K
,
L
,
J
,
I
K,L,J,I
K
,
L
,
J
,
I
respectively. Prove that if
D
I
=
C
F
DI=CF
D
I
=
CF
and
B
J
=
C
E
BJ=CE
B
J
=
CE
, then
K
J
=
I
L
KJ=IL
K
J
=
I
L
4
1
Hide problems
n! begins with 2005
Is there integer
n
n
n
such that
n
!
n!
n
!
begins with
2005
2005
2005
?