There are n positive integers. For each pair Boris writes their arithmetic mean on a blackboard and their geometric mean on a whiteboard. For every pair at least one of the two means is an integer. Prove that at least one board contains only integers.
Baron Munchausen claims: if the monic polynomial $x^n-ax^{n-1}+bx^{n-2}+\cdots$ has n positive integer roots, then there exist a lines in the plane having exactly b distinct intersection points. Is the claim true? The displayed coefficients presume $n\ge2$.
Two circles $\alpha,\beta$ with centers A,B intersect at C,D. Segment AB meets $\alpha$ at K and $\beta$ at L. Ray DK meets $\beta$ again at N, and ray DL meets $\alpha$ again at M. Prove that the intersection of the diagonals of quadrilateral KLMN is the incenter of triangle ABC.
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?
Alice and Bob alternate turns, with Alice first. Each written number is a unit fraction $1/n$ with n a positive integer. Alice writes one fraction on each turn; Bob writes k fractions on his kth turn. Bob wants the sum of all fractions written to be an integer after some turn. Can Alice prevent this forever?
A grid has 1000 columns and n rows, where n is odd and greater than 2020. A white bug starts in one corner; black chess bishops occupy the two nearest corners. The bug moves one cell horizontally or vertically, or by a knight’s move. It must reach the opposite corner, visiting every cell not occupied or attacked by a bishop exactly once and never visiting a forbidden cell. Prove that the number of such routes is independent of n.