MathDB

1

sum (n k)^2 = (2n n), binomials

{n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}

4

1

z_1^n + z_2^n is real for roots of z^2+az+b = 0

If

z_1

,

z_2

are the roots of the equation with real coefficients

z^2+az+b = 0

, prove that

z^n_1 + z^n_2

is a real number for any natural value of

n

. If particular of the equation

z^2 - 2z + 2 = 0

, express, as a function of

n

, the said sum.

2

1

transformation for reverse order of a written word

A certain Oxford professor, assigned to espionage cryptography services British, role played by Dirk Bogarde in a film, recruits his proposing small attention exercises, such as mentally reading a word the other way around. Frequently he does it with his own name:

SEBASTIAN

, what will there be to read

NAITSABES

. He wonders if there is any movement of the plane or of space that transforms one of these words in the other, just as they appear written. And if it had been called

AVITO

, like a certain Unamuno character? Give a reasoned explanation for each answer.

1

x^2/16 + y^2/ 4 =1 , (x -2)^2 + (y - 1)^2 = 5

Calculate the area of the intersection of the interior of the ellipse

\frac{x^2}{16}+ \frac{y^2}{4}= 1

with the circle bounded by the circumference

(x -2)^2 + (y - 1)^2 = 5

.

5

1

\int_2^4 \sin ((x-3)^3) dx

Calculate the definite integral

\int_2^4 \sin ((x-3)^3) dx

8

1

P^(325)(0) if P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}

Given the polynomial

P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.

Express the numerical value of its derivative of order

325

for

x = 0

.

7

1

volume of a tire

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

6

1

urn was filled with 3 balls , tossing a coin

An urn was filled with three balls by the following procedure: it was thrown a coin three times, inserting, each time a white ball came up heads, and every time tails came up, a black ball. We draw from this urn, four times consecutive, one ball; we return it to the urn before the next extraction. Which is the probability that in the four extractions a cue ball is obtained?

1979 Spain Mathematical Olympiad

Subcontests

sum (n k)^2 = (2n n), binomials

z_1^n + z_2^n is real for roots of z^2+az+b = 0

transformation for reverse order of a written word

x^2/16 + y^2/ 4 =1 , (x -2)^2 + (y - 1)^2 = 5

\int_2^4 \sin ((x-3)^3) dx

P^(325)(0) if P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}

volume of a tire

urn was filled with 3 balls , tossing a coin