2009 India Regional Mathematical Olympiad

Part of Regional Mathematical Olympiad

Subcontests

(6)

1

points on boundary of convex polygon

A convex polygon is such that the distance between any two vertices does not exceed

1

(i)

Prove that the distance between any two points on the boundary of the polygon does not exceed

1

(ii)

X

Y

are two distinct points inside the polygon, prove that there exists a point

Z

on the boundary of the polygon such that XZ \plus{} YZ\le1.

1

find number of pages torn

In a book with page numbers from

1

100

some pages are torn off. The sum of the numbers on the remaining pages is

4949

. How many pages are torn off?

1

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.

1

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

2009^{182}

1

a^2-3a-19 not divisible by 289

Show that there is no integer

a

such that a^2 \minus{} 3a \minus{} 19 is divisible by

289

1

Find angle in isosceles triangle ABC

ABC

be a triangle in which AB \equal{} AC and let

I

be its in-centre. Suppose BC \equal{} AB \plus{} AI. Find

\angle{BAC}