MathDB

1

sin^2 x - 2 cos^2 x +\frac12 sin 2x = 0

Calculate the values of the cosines of the angles

x

that satisfy the next equation:

\sin^2 x - 2 \cos^2 x +\frac12 \sin 2x = 0.

8

1

cos (AB,OM) wanted if sum of areas (AMC)+( BMD) is min

There is a point

M

inside a circle, at a distance

OM = d

of the center

O

. Two chords

AB

and

CD

are traced through

M

that form a right angle . Join

A

with

C

and

B

with

D

. Determine the cosine of the angle that must form the chord

AB

with

OM

so that the sum of the areas of the triangles

AMC

and

BMD

be minimal.

6

1

locus of circumcenters of AMN wanted, variable secant

Given a circle

\gamma

and two points

A

and

B

in its plane. By

B

passes a variable secant that intersects

\gamma

at two points

M

and

N

. Determine the locus of the centers of the circles circumscribed to the triangle

AMN

.

3

1

area bisector of triangle wanted passing through a random point on a side

An arbitrary triangle

ABC

is given and a point

P

lies on the side

AB

. It is requested to draw through

P

a line that divides the triangle into two figures of the same area.

5

1

percentages for 4 subjects in exams

In the sixth-year exams of a Center, they pass Physics at least

70\%

of the students, Mathematics at least

75\%

; Philosophy at least, the

90\%

and the Language at least,

85\%

. How many students, at least, pass these four subjects?

4

1

x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50, 2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25

Knowing that the polynomials

2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25

x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50

have two common double roots, determine all their roots.

2

1

a plant grows

A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the load of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the distance of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud. If n is the number of bifurcations that a certain plant of that type has, it is asks a) the number of branches of the plant, b) the number of buds, c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds.
Hint: You can proceed by induction, showing that if some results are correct for a given plant, they remain correct for the plant that is obtained substituting a bud in it for a pair of branches ending in individual buds.

1

cylindrical container of revolution is partially filled with a liquid

A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined

30^o

with respect to the vertical, we observe that when removing liquid so that the level falls

1

cm, the weight of the contents decreases

40

g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of

45^o

with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.

1970 Spain Mathematical Olympiad

Subcontests

sin^2 x - 2 cos^2 x +\frac12 sin 2x = 0

cos (AB,OM) wanted if sum of areas (AMC)+( BMD) is min

locus of circumcenters of AMN wanted, variable secant

area bisector of triangle wanted passing through a random point on a side

percentages for 4 subjects in exams

x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50, 2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25

a plant grows

cylindrical container of revolution is partially filled with a liquid