In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.

Let $a_1, a_2,\cdots , a_n$ and $b_1, b_2,\cdots , b_n$ be (not necessarily distinct) positive integers. We continue the sequences as follows: For every $i>n$, $a_i$ is the smallest positive integer which is not among $b_1, b_2,\cdots , b_{i-1}$, and $b_i$ is the smallest positive integer which is not among $a_1, a_2,\cdots , a_{i-1}$. Prove that there exists $N$ such that for every $i>N$ we have $a_i=b_i$ or for every $i>N$ we have $a_{i+1}=a_i$.

Find the least possible value for the fraction $$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$over all distinct positive integers $a, b, c$. By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.

Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties: There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$. There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$.