MathDB

1

sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...>9/2

\frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1 \cdot 2}+\sqrt[3]{2^2} }+\frac{1}{\sqrt[3]{3^2}+\sqrt[3]{3 \cdot 4}+\sqrt[3]{4^2} }+...+ \frac{1}{\sqrt[3]{999^2}+\sqrt[3]{999 \cdot 1000}+\sqrt[3]{1000^2} }> \frac{9}{2}

(The member on the left has 500 fractions.)

6

1

each guest receives two hats at a party

At a big New Year's Eve party, each guest receives two hats: one red and one blue. At the beginning of the party, all the guests put on the red hat. Several times throughout the evening, the announcer announces the name of one of the guests and, at that moment, the named and each of his friends change the hat they are wearing for the other color. Show that the announcer can make it so that all the guests are wearing the blue hat when the party is over.
Note: All guests remain at the party from start to finish.

3

1

coloring 111^2 points

Two players

A

and

B

play the following game:

A

chooses a point, with integer coordinates, on the plane and colors it green, then

B

chooses

10

points of integer coordinates, not yet colored, and colors them yellow. The game always continues with the same rules;

A

and

B

choose one and ten uncolored points and color them green and yellow, respectively. a. The objective of

A

is to achieve

111^2

green points that are the intersections of

111

horizontal lines and

111

vertical lines (parallel to the coordinate axes).

B

's goal is to stop him. Determine which of the two players has a strategy that ensures you achieve your goal. b. The objective of

A

is to achieve

4

green points that are the vertices of a square with sides parallel to the coordinate axes.

B

's goal is to stop him. Determine which of the two players has a strategy that will ensure that they achieve their goal.

2

1

for every number c, there exists n, such that a_{n+1} -a_n > c,

Let

p_1, p_2, ..., p_k

be

k

different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence:

a_1 < a_2 < ... < a_n < ...

Prove that, for every number

c

, there exists

n

such that

a_{n+1} -a_n > c

.

5

1

max of sum of distances of point of semicircle from 3 sides of cyclic ABCD

The quadrilateral

ABCD

is inscribed in a circle of radius

1

, so that

AB

is a diameter of the circumference and

CD = 1

. A variable point

X

moves along the semicircle determined by

AB

that does not contain

C

or

D

. Determine the position of

X

for which the sum of the distances from

X

to lines

BC, CD

and

DA

is maximum.

1

scent of a classic geometry

Let

ABC

be a scalene acute triangle whose orthocenter is

H

.

M

is the midpoint of segment

BC

.

N

is the point where the segment

AM

intersects the circle determined by

B, C

, and

H

. Show that lines

HN

and

AM

are perpendicular.

1999 Rioplatense Mathematical Olympiad, Level 3

Subcontests

sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...>9/2

each guest receives two hats at a party

coloring 111^2 points

for every number c, there exists n, such that a_{n+1} -a_n > c,

max of sum of distances of point of semicircle from 3 sides of cyclic ABCD

scent of a classic geometry

1999 Rioplatense Mathematical Olympiad, Level 3

Subcontests

sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...&gt;9/2

each guest receives two hats at a party

coloring 111^2 points

for every number c, there exists n, such that a_{n+1} -a_n &gt; c,

max of sum of distances of point of semicircle from 3 sides of cyclic ABCD

scent of a classic geometry

sum 1/ \sqrt[3]{1^2}+\sqrt[3]{1 x 2}+\sqrt[3]{2^2} } + ...>9/2

for every number c, there exists n, such that a_{n+1} -a_n > c,