MathDB

1

ax^3 + bx^2 + cx inflection point

a, b, c

, so that the graphical representation of the function

y = ax^3 + bx^2 + cx

has an inflection point at the point of abscissa

x = 3

, with tangent at the point of equation

x - 4y + 1 = 0.

Then draw the corresponding graph.

7

1

geometric progression of seven terms wanted

Determine a geometric progression of seven terms, knowing the sum,

7

, of the first three, and the sum,

112

, of the last three.

6

1

married couple has 5 children

They tell us that a married couple has

5

children. Calculate the probability that among them there are at least two men and at least one woman. Probability of being born male is considered

1/2

.

5

1

computational geo related to a right triangle

The length of the hypotenuse

BC

of a right triangle

ABC

is

a

, and on it the points

M

and

N

are taken such that

BM = NC = k

, with

k < a/2

. Assuming that (only) the data

a

and

k

are known, calculate: a) The value of the sum of the squares of the lengths

AM

and

AN

. b) The ratio of the areas of triangles

ABC

and

AMN

. c) The area enclosed by the circle that passes through the points

A, M' , N'

, where

M'

is the orthogonal projection of

M

onto

AC

and

N'

that of

N

onto

AB

.

4

1

hang a weight 7 m below a ceiling

You want to hang a weight

P

so that it is

7

m below a ceiling. To do this, it is suspended by means of a vertical cable attached to the midpoint

M

of a chain hung by its ends from two points on the ceiling

A

and

B

distant from each other

4

m. The price of the cable

PM

is

p

pta/m and that of the chain

AMB

is

q

pta/m. It is requested: a) Determine the lengths of the cable and the chain to obtain the lowest price cost of installation. b) Discuss the solution for the different values of the relation

p/q

of both prices. (It is assumed that the weight is large enough to be considered rectile lines the chain segments

AM

and

MB

).

3

1

convex pentagon drawn by diagonals of regular pentagon

Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

2

1

xyz in the base 7, zyx$ in the base 9 system

A three-digit number is written

xyz

in the base

7

system and

zyx

in the base

9

system . What is the number?

1

100 units split into 50,70, 65 as max as posible from 50

To a manufacturer of three products whose unit prices are

50

,

70

, and

65

pta, a retailer asks him for

100

units, remitting him

6850

pta as payment, on the condition that you send as many of the higher-priced product as possible and the rest of the other two. How many of each product should he send to serve the request?

1966 Spain Mathematical Olympiad

Subcontests

ax^3 + bx^2 + cx inflection point

geometric progression of seven terms wanted

married couple has 5 children

computational geo related to a right triangle

hang a weight 7 m below a ceiling

convex pentagon drawn by diagonals of regular pentagon

xyz in the base 7, zyx$ in the base 9 system

100 units split into 50,70, 65 as max as posible from 50