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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2011 Singapore Junior Math Olympiad
2011 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
5
1
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exactly one hundred copies of 1 on the board, replacing numbers with1,10,25
Initially, the number
10
10
10
is written on the board. In each subsequent moves, you can either (i) erase the number
1
1
1
and replace it with a
10
10
10
, or (ii) erase the number
10
10
10
and replace it with a
1
1
1
and a
25
25
25
or (iii) erase a
25
25
25
and replace it with two
10
10
10
. After sometime, you notice that there are exactly one hundred copies of
1
1
1
on the board. What is the least possible sum of all the numbers on the board at that moment?
4
1
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a_{m+i} is the odd part of 3a_m + 1m find a_{2011}
Any positive integer
n
n
n
can be written in the form
n
=
2
a
q
n = 2^aq
n
=
2
a
q
, where
a
≥
0
a \ge 0
a
≥
0
and
q
q
q
is odd. We call
q
q
q
the odd part of
n
n
n
. Define the sequence
a
0
,
a
1
,
.
.
.
a_0,a_1,...
a
0
,
a
1
,
...
as follows:
a
0
=
2
2011
−
1
a_0 = 2^{2011}-1
a
0
=
2
2011
−
1
and for
m
>
0
,
a
m
+
i
m > 0, a_{m+i}
m
>
0
,
a
m
+
i
is the odd part of
3
a
m
+
1
3a_m + 1
3
a
m
+
1
. Find
a
2011
a_{2011}
a
2011
.
1
1
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xy >= ac + bd if x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2},a,b,c,d> 0
Suppose
a
,
b
,
c
,
d
>
0
a,b,c,d> 0
a
,
b
,
c
,
d
>
0
and
x
=
a
2
+
b
2
,
y
=
c
2
+
d
2
x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2}
x
=
a
2
+
b
2
,
y
=
c
2
+
d
2
. Prove that
x
y
≥
a
c
+
b
d
xy \ge ac + bd
x
y
≥
a
c
+
b
d
.
2
1
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tangent circles, angle bisector wanted (Singapore Junior 2011)
Two circles
Γ
1
,
Γ
2
\Gamma_1, \Gamma_2
Γ
1
,
Γ
2
with radii
r
i
,
r
2
r_i, r_2
r
i
,
r
2
, respectively, touch internally at the point
P
P
P
. A tangent parallel to the diameter through
P
P
P
touches
Γ
1
\Gamma_1
Γ
1
at
R
R
R
and intersects
Γ
2
\Gamma_2
Γ
2
at
M
M
M
and
N
N
N
. Prove that
P
R
PR
PR
bisects
∠
M
P
N
\angle MPN
∠
MPN
.
3
1
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SMO Junior 2011 Rd 2 Q3
Let
S
1
,
S
2
,
.
.
.
S
2011
\text{Let} S_1,S_2,...S_{2011}
Let
S
1
,
S
2
,
...
S
2011
be nonempty sets of consecutive integers such that any
\text{be nonempty sets of consecutive integers such that any}
be nonempty sets of consecutive integers such that any
2
2
2
of them have a common element. Prove that there is a positive integer that belongs to every
\text{of them have a common element. Prove that there is a positive integer that belongs to every}
of them have a common element. Prove that there is a positive integer that belongs to every
S
i
,
i
=
1
,
.
.
.
,
2011
S_i, i=1,...,2011
S
i
,
i
=
1
,
...
,
2011
(For example,
2
,
3
,
4
,
5
{2,3,4,5}
2
,
3
,
4
,
5
is a set of consecutive integers while
2
,
3
,
5
{2,3,5}
2
,
3
,
5
is not.)