1990 India Regional Mathematical Olympiad

Part of Regional Mathematical Olympiad

Subcontests

(8)

1

Very simple!!

If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.

1

An old puzzle

A census man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said "We do not mind giving you the sum of the ages of any two ladies you may choose". Thereupon, the census man said, "In that case, please give me the sum of the ages of every possible pair of you". They gave the sums as: 30, 33, 41, 58, 66, 69. The census man took these figures and happily went away. How did he calculate the individual ages?

1

Find the digit

N

50

-digit number (in decimal representation). All digits except the

26

th digit (from the left) are

1

N

is divisible by

13

26

1

Triangle inequality

P

is any point inside a triangle

ABC

. The perimeter of the triangle

AB + BC + Ca = 2s

s < AP +BP +CP < 2s

1

Remainder?

Find the remainder when

2^{1990}

1990.

1

Folding...

A square sheet of paper

ABCD

is so folded that

B

falls on the mid point of

M

CD

. Prove that the crease will divide

BC

5 : 3

1

Simple inequality

For all positive real numbers

a,b,c

, prove that \frac {a}{b \plus{} c} \plus{} \frac {b}{c \plus{} a} \plus{} \frac {c}{a \plus{} b} \geq \frac {3}{2}.

1

Yaphp!!!

Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least two of them will always be of the same size(radius). Prove that there are at least three ball which lie in the same box have the same colour and have the same size(radius).