Does there exist a positive integer divisible by 2020 in which all ten digits 0,1,…,9 occur equally often? The book’s printed sentence only asks that the digits occur; the equal-frequency condition is present in the official original and is used here.
Three knights attack a dragon. The first removes half its current heads plus 1; the second removes one third plus 2; the third removes one quarter plus 3. Each attack must remove an integer number of existing heads. They may choose any order and repeat attacks; if no legal attack exists the dragon wins. Can they remove all heads when the initial count is $41!$?
Can an $N$-gon be inscribed in a circle so that all side lengths are distinct and all its interior angles are integers in degrees, for (a) $N=19$; (b) $N=20$?
For which positive integers $N$ can an $N\times N$ grid be filled with real numbers so that the sums over all side-adjacent pairs are exactly the integers 1 through $2N(N-1)$, each once? Include the degenerate one-cell case explicitly.
Alice divides a 36-card deck, with 4 suits and 9 ranks, into 18 cards for herself and 18 for Bob. Each round Alice plays one remaining card, then Bob plays one. Bob scores 1 if their cards have the same suit or rank. What score can Bob guarantee against every initial division and every play by Alice? The official original says the cards are face up; the book says face down. Use the official face-up rule for the intended problem.
Gleb chooses positive integers $a\lt N$ and writes $a$. Repeatedly divide fixed $N$ by the last positive number written and append the remainder; stop when 0 appears. Can the sum of all written numbers exceed $100N$? This is the official original rule. The book reverses the division, saying to divide the last number by $N$; that literal version is discussed separately below.