MathDB

4

Part of 2003 IMC

Problems(2)

IMC 2003 / A4

Source: IMC 2003 day 1 problem 4

7/29/2005
Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that a⋅A=b⋅Ba\cdot A=b\cdot B.
IMCcollege contests
IMC 2003 Problem 10

Source: IMC 2003 Day 2 Problem 4

11/2/2020
Find all the positive integers nn for which there exists a Family F\mathcal{F} of three-element subsets of S={1,2,...,n}S=\{1,2,...,n\} satisfying (i) for any two different elements a,b∈S there exists exactly one A∈F containing both a and b;\text{(i) for any two different elements $a,b \in S$ there exists exactly one $A \in \mathcal{F}$ containing both $a$ and $b$;} (ii) if a,b,c,x,y,z are elements of S such that {a,b,x},{a,c,y},{b,c,z}∈F, then {x,y,z}∈F .\text{(ii) if $a,b,c,x,y,z$ are elements of $S$ such that $\{a,b,x\},\{a,c,y\},\{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F} $ }.
combinatorics