CMOQR 2013

Determine all real solutions to the following equation: $$2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.$$

In triangle $ABC$ , $\angle A = 90^\circ$ and $\angle C = 70^\circ$ . $F$ is point on $AB$ such that $\angle ACF = 30^\circ$ , and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$ . Prove that $BE$ bisects $\angle B$ .

A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$ , $\text{lcm}(x, z) = 900$ , and $\text{lcm}(y, z) = n$ , where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.

Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that both of these events occur:

• (i) Each person receives exactly one gift;
• (ii) No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$ , then $B$ did not give her gift to $A$ ).