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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2009 India Regional Mathematical Olympiad
2009 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
5
1
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points on boundary of convex polygon
A convex polygon is such that the distance between any two vertices does not exceed
1
1
1
.
(
i
)
(i)
(
i
)
Prove that the distance between any two points on the boundary of the polygon does not exceed
1
1
1
.
(
i
i
)
(ii)
(
ii
)
If
X
X
X
and
Y
Y
Y
are two distinct points inside the polygon, prove that there exists a point
Z
Z
Z
on the boundary of the polygon such that XZ \plus{} YZ\le1.
6
1
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find number of pages torn
In a book with page numbers from
1
1
1
to
100
100
100
some pages are torn off. The sum of the numbers on the remaining pages is
4949
4949
4949
. How many pages are torn off?
4
1
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Sum of 3-digit nos. with atleast 1 odd and 1 even digit
Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
3
1
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factorise 3^2008+4^2009
Show that 3^{2008} \plus{} 4^{2009} can be written as product of two positive integers each of which is larger than
200
9
182
2009^{182}
200
9
182
.
2
1
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a^2-3a-19 not divisible by 289
Show that there is no integer
a
a
a
such that a^2 \minus{} 3a \minus{} 19 is divisible by
289
289
289
.
1
1
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Find angle in isosceles triangle ABC
Let
A
B
C
ABC
A
BC
be a triangle in which AB \equal{} AC and let
I
I
I
be its in-centre. Suppose BC \equal{} AB \plus{} AI. Find
∠
B
A
C
\angle{BAC}
∠
B
A
C