MathDB

1

sum n /3 57...(2n+1) <1/2 - 1968 Polish Mo finals p2

n

\frac{1}{3} + \frac{2}{3\cdot 5} + \frac{3}{3 \cdot 5 \cdot 7} + ...+ \frac{n}{3 \cdot 5 \cdot 7 \cdot ...\cdot (2n+1)} < \frac{1}{2}.

3

1

6 points lie on surface of sphere

In a tetrahedron

ABCD

the edges

AD

,

BD

,

CD

are equal.

ABC

Non-collinear points are chosen in the plane.

A_1

,

B_1

,

C_1

The lines

DA_1

,

DB_1

,

DC_1

intersect the surface of the sphere circumscribed about the tetrahedron at points

A_2

,

B_2

,

C_2

, different from the point

D

. Prove that the points

A_1

,

B_1

,

C_1

,

A_2

,

B_2

,

C_2

lie on the surface of a certain sphere.

1

n pairs of parallel lines divides plane

What is the largest number of regions into which a plane can be divided by drawing

n

pairs of parallel lines?

4

1

pairwise sums a_i + a_j contains as few different numbers as possible

Given an integer

n > 2

, give an example of a set of

n

mutually different numbers

a_1,...,a_n

for which the set of their pairwise sums

a_i + a_j

(

i \ne j

) contains as few different numbers as possible; also give an example of a set of n different numbers

b_1,...,b_n

for which the set of their pairwise sums

b_i+b_j

(

i \ne j

) contains as many different numbers as possible;

6

1

n points

Consider a set of

n > 3

points in the plane, no three of which are collinear, and a natural number

k < n

. Prove the following statements:
(a) If

k \le \frac{n}{2}

, then each point can be connected with at least k other points by segments so that no three segments form a triangle.
(b) If

k \ge \frac{n}{2}

, and each point is connected with at least k other points by segments, then some three segments form a triangle.

5

1

any four of b points are vertices of a convex quadrilateral,

Given

n \ge 4

points in the plane such that any four of them are the vertices of a convex quadrilateral, prove that these points are the vertices of a convex polygon.

1968 Polish MO Finals

Subcontests

sum n /3 57...(2n+1) <1/2 - 1968 Polish Mo finals p2

6 points lie on surface of sphere

n pairs of parallel lines divides plane

pairwise sums a_i + a_j contains as few different numbers as possible

n points

any four of b points are vertices of a convex quadrilateral,

1968 Polish MO Finals

Subcontests

sum n /3 *5*7*...*(2n+1) &lt;1/2 - 1968 Polish Mo finals p2

6 points lie on surface of sphere

n pairs of parallel lines divides plane

pairwise sums a_i + a_j contains as few different numbers as possible

n points

any four of b points are vertices of a convex quadrilateral,

sum n /3 57...(2n+1) <1/2 - 1968 Polish Mo finals p2