MathDB

1

Indian Mathematical Olympiad 1990 - Problem 5

a

,

b

,

c

denote the sides of a triangle. Show that the quantity \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b} must lie between the limits

3/2

and 2. Can equality hold at either limits?

3

1

Indian Mathematical Olympiad 1990 - Problem 3

Let

f

be a function defined on the set of non-negative integers and taking values in the same set. Given that (a) \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right] for all non-negative integers

x

; (b)

1900 < f(1990) < 2000

, find the possible values that

f(1990)

can take. (Notation : here

[z]

refers to largest integer that is

\leq z

, e.g. [3.1415] \equal{} 3).

1

Indian Mathematical Olympiad 1990 - Problem 1

Given the equation x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0 has four real, positive roots, prove that (a) pr \minus{} 16s \geq 0 (b) q^2 \minus{} 36s \geq 0 with equality in each case holding if and only if the four roots are equal.

2

1

Indian Mathematical Olympiad 1990 - Problem 2

Determine all non-negative integral pairs

(x, y)

for which (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.

4

1

Indian Mathematical Olympiad 1990 - Problem 4

Consider the collection of all three-element subsets drawn from the set

\{1,2,3,4,\dots,299,300\}

. Determine the number of those subsets for which the sum of the elements is a multiple of 3.

7

1

Indian Mathematical Olympiad 1990 - Problem 7

Let

ABC

be an arbitrary acute angled triangle. For any point

P

lying within the triangle, let

D

,

E

,

F

denote the feet of the perpendiculars from

P

onto the sides

AB

,

BC

,

CA

respectively. Determine the set of all possible positions of the point

P

for which the triangle

DEF

is isosceles. For which position of

P

will the triangle

DEF

become equilateral?

6

1

Indian Mathematical Olympiad 1990 - Problem 6

Triangle

ABC

is scalene with angle

A

having a measure greater than 90 degrees. Determine the set of points

D

that lie on the extended line

BC

, for which |AD|\equal{}\sqrt{|BD| \cdot |CD|} where

|BD|

refers to the (positive) distance between

B

and

D

.

1990 India National Olympiad

Subcontests

Indian Mathematical Olympiad 1990 - Problem 5

Indian Mathematical Olympiad 1990 - Problem 3

Indian Mathematical Olympiad 1990 - Problem 1

Indian Mathematical Olympiad 1990 - Problem 2

Indian Mathematical Olympiad 1990 - Problem 4

Indian Mathematical Olympiad 1990 - Problem 7

Indian Mathematical Olympiad 1990 - Problem 6