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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2001 Singapore Senior Math Olympiad
2001 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(3)
1
1
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sum sin x_i=0, sum n sin x_i=100, system
Let
n
n
n
be a positive integer. Suppose that the following simultaneous equations
{
sin
x
1
+
sin
x
2
+
.
.
.
+
sin
x
n
=
0
sin
x
1
+
2
sin
x
2
+
.
.
.
+
n
sin
x
n
=
100
\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\ \sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases}
{
sin
x
1
+
sin
x
2
+
...
+
sin
x
n
=
0
sin
x
1
+
2
sin
x
2
+
...
+
n
sin
x
n
=
100
has a solution, where
x
1
x
2
,
.
.
,
x
n
x_1 x_2,.., x_n
x
1
x
2
,
..
,
x
n
are the unknowns. Find the smallest possible positive integer
n
n
n
. Justify your answer.
2
1
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min f(n+2)/f(n), if f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1
Let
n
n
n
be a positive integer, and let
f
(
n
)
=
1
n
+
2
n
−
1
+
3
n
−
2
+
4
n
−
3
+
.
.
.
+
(
n
−
1
)
2
+
n
1
f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1
f
(
n
)
=
1
n
+
2
n
−
1
+
3
n
−
2
+
4
n
−
3
+
...
+
(
n
−
1
)
2
+
n
1
Find the smallest possible value of
f
(
n
+
2
)
f
(
n
)
\frac{f(n+2)}{f(n)}
f
(
n
)
f
(
n
+
2
)
.Justify your answer.
3
1
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1 to 50 insides a 50x50 square
Each of the squares in a
50
×
50
50 \times 50
50
×
50
square board is filled with a number from
1
1
1
to
50
50
50
so that each of the numbers
1
,
2
,
.
.
.
,
50
1,2, ..., 50
1
,
2
,
...
,
50
appears exactly
50
50
50
times. Prove that there is a row or a column containing at least
8
8
8
distinct numbers.