MathDB

2

Part of 2012 Serbia National Math Olympiad

Problems(2)

Find all (a,b) s.t. a|b^2, b|a^2, a+1 | b^2 + 1

Source: Serbian National Olympiad 2012, Problem 2

4/4/2012
Find all natural numbers aa and bb such that a|b^2,   b|a^2 \mbox{ and } a+1|b^2+1.
number theorygreatest common divisoralgebrapolynomialVietainequalitiesnumber theory proposed
Bijection from natural numbers to integer lattice

Source: Serbian National Olympiad 2012, Problem 5

4/4/2012
Let K\mathbb{K} be two-dimensional integer lattice. Is there a bijection f:N→Kf:\mathbb{N} \rightarrow \mathbb{K}, such that for every distinct a,b,c∈Na,b,c \in \mathbb{N} we have: \gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \mbox{ are not colinear? }
combinatorics proposedcombinatorics