Question

[Carthrine]

Tìm tất cả các số nguyên tố a,b,c (có thể bằng nhau) thỏa mãn:

abc<ab+bc+ca

Answer 1

[Nguyen Viet Dat]

Chia hai ve cho cua bat dang thuc abc<ab+bc+ca cho abc ta duoc 1<\(\frac{1}{c}+\frac{1}{a}+\frac{1}{b}\)(3)

Vi a,b,c la SNT \(\Rightarrow a;b;c\ge2\)(1)

Gia su a>b>c . Trong ba phan so \(\frac{1}{c};\frac{1}{a};\frac{1}{b}\)thi \(\frac{1}{c}\)la phan so lon nhat nen \(\frac{1}{c}>\frac{1}{3}\) hay c<3(2)

Tu (1) va (2) suy ra c=2

Thay c=2 vao (3) ta duoc \(\frac{1}{a}+\frac{1}{b}<\frac{1}{2}\)(4)

Trong hai phan so \(\frac{1}{a};\frac{1}{b}\),phan so \(\frac{1}{b}\) lon hon nen \(\frac{1}{b}>\frac{1}{2}:2=\frac{1}{4}\Rightarrow b<4\) ma b>c=2 vay b=3

Thay b=3 vao (4) ta duoc \(\frac{1}{a}>\frac{1}{6}\) .Do do a<6 ma a>b=3 va a la so nguyen to vay a=5

Vay cap so a;b;c thoa mam la:(2;3;5);(2;5;3);(3;5;2);(3;2;5);(5;3;2);(5;2;3)

Answer 2

[Nguyen Viet Dat]

khong co so nao thoa man