MathDB

2

Part of 2011 Pre - Vietnam Mathematical Olympiad

Problems(2)

Ineq with the number of elements in the sets

Source: Pre-VMO 2012 - Problem 2

11/27/2011
Let AA be a set of finite distinct positive real numbers. Two other sets BB, CC are defined by: B={xy;x,y∈A},      C={xy;x,y∈A}B = \left\{ {\frac{x}{y};x,y \in A} \right\},\; \; \; C = \left\{ {xy;x,y \in A} \right\} Prove that ∣A∣.∣B∣≤∣C∣2\left| A \right|.\left| B \right| \le {\left| C \right|^2}.
combinatorics proposedcombinatorics
System of functional equations

Source: Pre-VMO 2012 - Problem 6

11/27/2011
Find all function f,g:Q→Qf,g: \mathbb{Q} \to \mathbb{Q} such that f(g(x)−g(y))=f(g(x))−yg(f(x)−f(y))=g(f(x))−y\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array} for all x,y∈Qx,y \in \mathbb{Q}.
functionalgebra proposedalgebra