AT-OEMO-FED1 1989

Natural numbers $a \le b \le c \le d$ satisfy $a+b+c+d=30$ . Find the maximum value of the product $P=abcd.$

If $a$ and $b$ are nonnegative real numbers with $a^2+b^2=4$ , show that: $\frac{ab}{a+b+2} \le \sqrt{2}-1$ and determine when equality occurs.

Let $a$ be a real number. Prove that if the equation $x^2-ax+a=0$ has two real roots $x_1$ and $x_2$ , then: $x_1^2+x_2^2 \ge 2(x_1+x_2).$

Prove that for any triangle each exradius is less than four times the circumradius.