500 Solutions

No problem is ever permanently closed. The editor is always pleased to consider for publication new solutions or new insights o n p a s t p r oblems. (1) Determine the set of all c > 0 such that whenever (1) holds, then we have x + y + z xy + yz+ zx. The positive r eals c asked for are those satisfying c 4. From (1), we have xy < c and z = c ; xy x + y + xy. Substituting this value of z into the given inequality, we see that we must determine those positive r eals c such that which simpliies to A(xx y) : = (x + y) 2 ; (xy) 2 + (c ; xy)(1 ; x ; y) 0 , (2) whenever x, y > 0 such that x y < c. Taking x = c=2 and letting y approach zero, we obtain c 2 4 + c 1 ; c 2 0 , which means that c 4. Now suppose that c 4. Let x, y > 0 such that xy < c.