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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2010 Singapore Junior Math Olympiad
2010 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
4
1
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m/n =0.167a_1a_2... where n <=100 is wrong
A student divides an integer
m
m
m
by a positive integer
n
n
n
, where
n
≤
100
n \le 100
n
≤
100
, and claims that
m
n
=
0.167
a
1
a
2
.
.
.
\frac{m}{n}=0.167a_1a_2...
n
m
=
0.167
a
1
a
2
...
. Show the student must be wrong.
3
1
Hide problems
a_i + a_j = a_k + a_{\ell} between n positiv e integers, with 5 distinct at leas
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any
1
≤
i
<
j
≤
n
1 \le i < j \le n
1
≤
i
<
j
≤
n
, there exist
k
,
ℓ
k,\ell
k
,
ℓ
, both different from
i
i
i
and
j
j
j
such that
a
i
+
a
j
=
a
k
+
a
ℓ
a_i + a_j = a_k + a_{\ell}
a
i
+
a
j
=
a
k
+
a
ℓ
. What is the smallest possible value of
n
n
n
?
2
1
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5-digit integers not multiples of 11 and whose digits are 1, 3, 4, 7,
Find the sum of all the
5
5
5
-digit integers which are not multiples of
11
11
11
and whose digits are
1
,
3
,
4
,
7
,
9
1, 3, 4, 7, 9
1
,
3
,
4
,
7
,
9
.
1
1
Hide problems
inradius of PMSN equals to MP-MS (Singapore Junior 2010)
Let the diagonals of the square
A
B
C
D
ABCD
A
BC
D
intersect at
S
S
S
and let
P
P
P
be the midpoint of
A
B
AB
A
B
. Let
M
M
M
be the intersection of
A
C
AC
A
C
and
P
D
PD
P
D
and
N
N
N
the intersection of
B
D
BD
B
D
and
P
C
PC
PC
. A circle is incribed in the quadrilateral
P
M
S
N
PMSN
PMSN
. Prove that the radius of the circle is
M
P
−
M
S
MP- MS
MP
−
MS
.
5
1
Hide problems
SMO 2010 senior q2
The numbers
1
1
,
1
2
,
.
.
.
,
1
2010
\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}
1
1
,
2
1
,
...
,
2010
1
are written on a blackboard. A student chooses any two of the numbers, say
x
x
x
,
y
y
y
, erases them and then writes down
x
+
y
+
x
y
x + y + xy
x
+
y
+
x
y
. He continues to do this until only one number is left on the blackboard. What is this number?