If x and y are positive real numbers and m, n are any positive integers, then $${{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(1 + {y^{2m}})}} > {1 \over 4}$$

If $${n_1}$$, $${n_2}$$,.......$${n_p}$$ are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.

If $$P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c$$, where $$ac \ne \,0$$, then P(x) Q(x) = 0 has at least two real roots.

If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.