ITOT 41 Spring 2020 - THPT O - Additive practice

Can a 40-by 41 table be filled with one integer per cell so that each integer equals the number of side-adjacent cells containing that same integer?

After visiting Addis Ababa, Alice says she has celebrated New Year’s Eve inside every possible hemisphere of Earth except one. What is the least number of places where she could have celebrated it? Treat places as points on a sphere; a point on a hemisphere’s boundary is not inside. (The official English source calls the traveller Sasha.)

A circle carries 41 letters, each A or B. One may replace a consecutive ABA by B or reverse that replacement, and likewise replace BAB by A or reverse it. Can every initial circular word be reduced to a single letter?

Does a nonconstant real polynomial $p(x)$ exist that is representable as a sum $a(x)+b(x)$ of two squared real polynomials in (a) exactly one way; (b) exactly two ways? Interchanging the two summands does not give a different way. Original weights: 2+3.

Two circles intersect at $P,Q$. A line $\ell$ through $Q$ meets them again at $A,B$. Tangents at $A,B$ meet at $C$. The bisector of $\angle CPQ$ meets line $AB$ at $D$. Prove that all possible positions of $D$, as $\ell$ varies, lie on a single circle.