DE-MO 1999

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$ , when $x,y$ are a) natural numbers, b) integers

Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}$

A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$ : $A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$ . However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.

A convex polygon $P$ is placed inside a unit square $Q$ . Prove that the perimeter of $P$ does not exceed $4$ .

Consider the following inequality for real numbers $x,y,z$ : $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ . (a) Prove that the inequality is valid for $a = 2\sqrt2$ (b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$ .