If ~$r$~ and ~$s$~ are positive integers, is  ~$r$~ + ~$s$~ even?

1. ~$r$~ is even.

2. ~$s$~ is even. 

In the figure above,~$\overline{AB}$~, which has length ~$z$~cm, is tangent to the circle at point ~$A$~, and ~$\overline{BD}$~, which has length ~$y$~cm, intersects the circle at point ~$C$~. If ~$BC=x$~cm and ~$z=\sqrt{xy}$~, what is the value of ~$x$~?

(1) ~$CD=x$~cm

(2) ~$z=5\sqrt2$~

If ~$a, b$~, and ~$c$~ are consecutive integers and ~$0 8?

(1) The product ~$ac$~ is even.

(2)The product ~$bc$~ is a multiple of 4.

Last year ~$\frac{3}{5}$~ of the members of a certain club were males. This year the members of the club include all the members from last year plus some new members. Is the fraction of the members of the club who are males greater this year than last year?

(1)More than half of the new members are male.

(2) The number of members of the club this yearis ~$\frac{6}{5}$~  the number of members last year.

~$A$~ triangle has side length of ~$a, b$~, and ~$c$~ centimeters. Does each angle in the triangle measure less than 90 degrees?

(1) The 3 semicircles whose diametersare the sides of the triangle have areas that are equal to ~$3cm^2,4cm^2$~, and ~$6cm^2$~,respectively.

(2) ~$c

If ,what is the value of x?

(1)  ~$2x=(\frac{x}{2})^3$~

(2)  ~$x=2^{x-2}$~

what is the value of ~$x^2-y^2$~?

(1)  ~$(x-y)^2=9$~

(2) ~$x+y=6$~

If ~$N$~ is positive odd integer, is ~$N$~ prime?

(1) ~$N =2^k+1$~ for some positive integer ~$k$~.

(2) ~$N $~= 2 and ~$N$~ +4 are both prime.

If ~$n$~ is the least of three differentintegers greater than 1, what is the value of ~$n$~?

(1) The product of the three integers is90.

(2) One of the integers is twice one ifthe other two integers.

 If ~$a$~ and ~$b$~ are positive integers, what is the value of the product ~$ab$~?

(1) The least common multiple of ~$a$~ and ~$b$~ is 48.

(2)The greatest common factor of ~$a$~ and ~$b$~ is 4.