2011/042) Given $x^2+y^2 = 1$ and $r^2 + s^2 =1$ maximise $rx + sy$

solution:
this can be done by AM GM inequality also but here I show a different way

as $x^2+y^2 = 1$so let $x = \sin\, t , y= \cos\, t$
and $r^2+s^2 = 1$ so let $r = \sin\, a , s = \cos\, a$

$xr+ sy = \sin\, t \sin\, a+ \cos\, t \cos\, a = cos (t-a)$

maximum when $t = a$ that is 1 ( it is possible when x = r = 1 and y = s = 0)
so maximum 1 is possible
so 1 is the result