6

Part of 1997 IMC

Problems(2)

Familiy of finite subsets (sort of set theory)

Source: IMC 1997 day 1 problem 6

F

is a family of finite subsets of

\mathbb{N}

and for any 2 sets

A,B \in F

we have A \cap B \not= \O. (a) Is it true that there is a finite subset

Y

\mathbb{N}

such that for any

A,B \in F

we have A\cap B\cap Y \not= \O? (b) Is the above true if we assume that all members of

F

have the same size?

inductioncombinatorics solvedcombinatorics

cross the axis uncountably often

Source: IMC 1997 day 2 problem 6

f: [0,1]\rightarrow \mathbb{R}

continuous. We say that

f

crosses the axis at

x

f(x)=0

\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)

\epsilon

. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?

functiontrigonometryreal analysisreal analysis unsolved