Two Olympiad Problems

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straightforward solution to 2)

write

K = \sqrt{s(s-a)(s-b)(s-c)}

by heron formula,

substitute a=x+y,b=x+z,c=y+z for positive x,y,z (always possible in triangle). heron's formula becomes k = \sqrt{(x+y+z)(xyz)}

now it is straightforward to verify that (x+y+z)^2 \geq 3^{3/2} \sqrt{(x+y+z)(xyz)} = 3^{3/2}*K, with equality iff x=y=z.

in particular, we see that in an arbitrary triangle, 1/4 * P^2 \geq 3^{3/2} * K, with equality iff P is equilateral.