MathDB

1

partition of N in 3 subsets, a_1 < a_2 < ... < a_k and a_{j+1} -a_j \le m

N

is partitioned into three subsets

A_1,A_2,A_3

. Prove that at least one of them has the following property: There exists a positive number

m

such that for any

k

one can find numbers

a_1 < a_2 < ... < a_k

in that subset satisfying

a_{j+1} -a_j \le m

for

j = 1,...,k -1

.

5

1

in a group of mathematicians everybody has at least one friend

In a group of mathematicians everybody has at least one friend (friendship is a symmetric relation). Show that there is a mathematician all of whose friends have average number of friends not smaller than the average number of friends in the whole group.

4

1

distances in triangle with <A=2<B

Prove that in all triangles

ABC

with

\angle A = 2\angle B

the distance from

C

to

A

and to the perpendicular bisector of

AB

are in the same ratio.

3

1

permutations inequality

For any permutation

p

of the set

\{1,2,...,n\}

, let us denote

d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|

. Let

i(p)

be the number of inversions of

p

, i.e. the number of pairs

1 \le i < j \le n

with

p(i) > p(j)

. Prove that

d(p)\le 2i(p)

$.

2

1

n custodians in a 3n-gon shape museum

A museum has the shape of a (not necessarily convex) 3

n

-gon. Prove that

n

custodians can be positioned so as to control all of the museum’s space.

1

cos^2&phi;+q \sin&phi; \cos&phi; +r\sin^2&phi; \ge 1/2 (p+r-\sqrt{(p-r)^2+q^2})

Prove that for any real numbers

p,q,r,\phi

,:

\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})

1991 Czech And Slovak Olympiad IIIA

Subcontests

partition of N in 3 subsets, a_1 < a_2 < ... < a_k and a_{j+1} -a_j \le m

in a group of mathematicians everybody has at least one friend

distances in triangle with <A=2<B

permutations inequality

n custodians in a 3n-gon shape museum

cos^2&phi;+q \sin&phi; \cos&phi; +r\sin^2&phi; \ge 1/2 (p+r-\sqrt{(p-r)^2+q^2})

1991 Czech And Slovak Olympiad IIIA

Subcontests

partition of N in 3 subsets, a_1 &lt; a_2 &lt; ... &lt; a_k and a_{j+1} -a_j \le m

in a group of mathematicians everybody has at least one friend

distances in triangle with &lt;A=2&lt;B

permutations inequality

n custodians in a 3n-gon shape museum

cos^2&amp;phi;+q \sin&amp;phi; \cos&amp;phi; +r\sin^2&amp;phi; \ge 1/2 (p+r-\sqrt{(p-r)^2+q^2})

partition of N in 3 subsets, a_1 < a_2 < ... < a_k and a_{j+1} -a_j \le m

distances in triangle with <A=2<B

cos^2φ+q \sinφ \cosφ +r\sin^2φ \ge 1/2 (p+r-\sqrt{(p-r)^2+q^2})