ITOT 41 Spring 2020 - THPT A - Additive practice

Let $U$ be the closed region between $y=x^2$ and $y=x^2-1$. Does $U$ contain a straight line segment longer than $10^6$?

Alice chooses positive integers $a,b,c$. There is exactly one ordered triple of positive integers $(x,y,z)$ satisfying $a=\operatorname{lcm}(x,y)$, $b=\operatorname{lcm}(x,z)$, $c=\operatorname{lcm}(y,z)$. She tells Bob that the triple exists uniquely and gives him $a,b$. Prove that Bob can determine $c$.

Can a tetrahedron have two plane cross-sections that are squares, one with side at most 1 and the other with side at least 100?

Henry has 2 N distinct guests and N identical white hats and N identical black hats. He distributes all hats and arranges the guests into one or more circles, each containing at least two people, with adjacent hat colors different. Prove the number of arrangements is $(2N)!$. Circular order is taken in the usual oriented sense: rotations are identical, reversal is not identified.

Let $ABCD$ be cyclic. The circles with diameters $AB,CD$ intersect at $X_1,Y_1$; those with diameters $BC,AD$ at $X_2,Y_2$; and those with diameters $AC,BD$ at $X_3,Y_3$. Prove the lines $X_1Y_1,X_2Y_2,X_3Y_3$ have a common point. The stated intersections are assumed to be two distinct points for each pair.

There are 2 n consecutive integers on a board. Partition them into pairs and simultaneously replace each pair by its sum and one of its two signed differences. Prove that the resulting 2 n integers cannot again be consecutive.

An infinite square grid is initially white. For which positive integers k can finitely many, but at least one, cells be colored black so that every horizontal, vertical and either-direction diagonal line of cells contains exactly k black cells or none?