1991 India Regional Mathematical Olympiad

Part of Regional Mathematical Olympiad

Subcontests

(7)

1

Equal squares

64

8 \times 8

chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. Show that in fact all the

64

entries are equal.

1

Find all inetger s.t....

Find all integer values of

a

such that the quadratic expression

(x+a)(x+1991) +1

can be factored as a product

(x+b)(x+c)

b,c

1

Parallel lines

P_1

BC

ABC

and draw the following chain of lines:

P_1P_2

AC

P_2P_3

BC

P_3P_4

AB

P_4P_5

CA

P_5P_6

BC

P_2,P_5

AB

P_3,P_6

CA

P_4

BC

P_6P_1

AB

1

One on urns

There are two urns each containing an arbitrary number of balls. We are allowed two types of operations: (a) remove an equal number of balls simultaneously from both the urns, (b) double the number of balls in any of them Show that after performing these operations finitely many times, both the urns can be made empty.

1

Find the number

A four-digit number has the following properties: (a) It is a perfect square; (b) Its first two digits are equal (c) Its last two digits are equal. Find all such four-digit numbers.

1

A very simple one!!!!

a,b,c,d

be any four positive real numbers, then prove that

\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4.

1

Another triangle

P

be an interior point of a triangle

ABC

AP,BP,CP

BC,CA,AB

D,E,F

respectively. Show that

\frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}.

[hide="Remark"]This is known as Van Aubel's Theorem.