MathDB

equipartitionable family of finite sets

Source: Romania BMO TST 1989 p4

2/17/2020

A family of finite sets

\left\{ A_{1},A_{2},.......,A_{m}\right\}

is called equipartitionable if there is a function

\varphi:\cup_{i=1}^{m}

\rightarrow\left\{ -1,1\right\}

such that

\sum_{x\in A_{i}}\varphi\left(x\right)=0

for every

i=1,.....,m.

Let

f\left(n\right)

denote the smallest possible number of

n

-element sets which form a non-equipartitionable family. Prove that a)

f(4k +2) = 3

for each nonnegative integer

k

, b)

f\left(2n\right)\leq1+m d\left(n\right)

, where

m d\left(n\right)

denotes the least positive non-divisor of

n.

functionSetsdivisorcombinatorics

{A \choose r} denote the family of all r-element subsets of A, function

Source: Romania IMO TST 1989 1.4

2/17/2020

Let

r,n

be positive integers. For a set

A

, let

{A \choose r}

denote the family of all

r

-element subsets of

A

. Prove that if

A

is infinite and

f : {A \choose r} \to {1,2,...,n}

is any function, then there exists an infinite subset

B

of

A

such that

f(X) = f(Y)

for all

X,Y \in {B \choose r}

.

functioncombinatorics

sphere S remains tangent to a fixed sphere

Source: Romania IMO TST 1989 2.4

2/17/2020

Let

A,B,C

be variable points on edges

OX,OY,OZ

of a trihedral angle

OXYZ

, respectively. Let

OA = a, OB = b, OC = c

and

R

be the radius of the circumsphere

S

of

OABC

. Prove that if points

A,B,C

vary so that

a+b+c = R+l

, then the sphere

S

remains tangent to a fixed sphere.

spherefixedtangent spheres3D geometrytrihedral anglegeometry

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