ITOT 42 Autumn 2020 - THCS A - Additive practice

A circle cuts a quadrilateral nicely if it intersects each side at exactly two distinct points in the interior of that side. Does every convex quadrilateral admit such a circle?

A pair of distinct positive integers is lovely if both its arithmetic and geometric means are integers. Does every lovely pair have another lovely pair with the same arithmetic mean? Pairs $(a,b)$ and $(b,a)$ are regarded as identical.

On each turn Alice announces an integer $x$. Bob writes either $x$ or $x$ plus the sum of all numbers already on the board. Initially the board is empty. Can Alice force, after finitely many turns, (a) at least 100 occurrences of 5; (b) at least 100 occurrences of 10?

An X-pentomino consists of five unit squares: one central square and one attached along each of its four sides. Can nine such shapes be cut from an 8-by 8 square? Cuts need not follow its grid lines. The source picture only illustrates three shapes.

Do there exist 100 distinct positive integers such that the cube of one equals the sum of the cubes of the other 99?

Two round tables each have $n$ dwarves. Each dwarf is initially friends only with the two neighbours at their own table. A wizard may add 2 n friendships between pairs of dwarves, but a witch then removes n of these new friendships. For which $n$ can the wizard guarantee a seating of all 2 n dwarves at one round table with every adjacent pair friends?

A convex quadrilateral has the property that no three of its side lengths can form a nondegenerate triangle. Prove that at least one of its angles is (a) at most 60 degrees; (b) at least 120 degrees.