1975 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

f(1/2)=1975/ 2^{1975}

On the interval

[0,1]

are given functions

S(x) = 1 - x

T(x) = x/2

. Does there exist a function of the form

f = g_1\circ g_2\circ ... \circ g_n

n \in N

g_k

S(x)

T(x)

f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?

1

R/2r >= 1/ sin a/2 (1- sin a/2)

Show that it is possible to circumscribe a circle of radius

R

about, and inscribe a circle of radius

r

in some triangle with one angle equal to

a

, if and only if

\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}

1

\sum_{i=N}^{N+k} a_i \ge 0

(a_k)_{k=1}^{\infty}

has the property that there is a natural number

n

a_1 + a_2 +...+ a_n = 0

a_{n+k} = a_k

k

. Prove that there exists a natural number

N

\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...

1

segments on surface of regular tetrahedron

On the surface of a regular tetrahedron of edge length

1

are given finitely many segments such that every two vertices of the tetrahedron can be joined by a polygonal line consisting of given segments. Can the sum of the lengths of the given segments be less than

1+\sqrt3

1

digits are 1,3,7, and 9

All decimal digits of some natural number are

1,3,7

9

. Prove that one can rearrange its digits so as to obtain a number divisible by

7

1

find the minimum value

0<u<1

\alpha > 0

minimum such that there exists

\beta > 0

(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1