1) Prove if P(X, Y|Z) = P(X|Z)P(Y|Z) then P(X|Y, Z) = P(X|Z) # conditional independence
2) integral 1/x^2 dx from -1 to 1
3) lim x->0 sqrt(x^2)/x
4) The coupon collection problem.
5) The secretary problem
Refer to wikipedia for problem description of secretary and coupon collection. For 1) you will find the proof on wikipedia. Actually you can prove if and only if for statement 1.