Question

Cho a, b, c là 3 số nguyên a; b; c thỏa mãn ab + bc + ca = 1

Chứng minh: (a² + 1)(b² + 1)(c² + 1) là số chính phương

Answer 1

\(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\)

\(=\left(a^2+ab+bc+ac\right)\left(b^2+ab+bc+ac\right)\left(c^2+ab+bc+ac\right)\)

\(=\left[a\left(a+b\right)+c\left(a+b\right)\right]\left[b\left(a+b\right)+c\left(a+b\right)\right]\left[c\left(b+c\right)+a\left(b+c\right)\right]\)

\(=\left(a+c\right)\left(a+b\right)\left(a+b\right)\left(b+c\right)\left(a+c\right)\left(b+c\right)\)

\(=\left[\left(a+b\right)\left(b+c\right)\left(a+c\right)\right]^2\rightarrow scp\)