There are $N$ untidy employees in an office, each with a pile of rubbish on their desk. They leave for lunch one at a time, each only after the preceding employee returns. Whenever an employee leaves, every other employee puts half of the rubbish on their own desk onto the absent employee’s desk. Is it possible that, after everyone has had lunch, every desk has exactly its initial amount of rubbish, if (a) $N=2$; (b) $N=10$? Original weights: (a) 1 point, (b) 3 points.
There are 2023 dice on a table. For one dollar, one die may be turned so that any of its four side faces becomes the top face. What is the smallest number of dollars sufficient to guarantee making the top numbers on all dice equal? The faces are numbered 1 through 6 and opposite faces always sum to 7.
For any real $x$, define $Q(x)=\lfloor x\rfloor+\lfloor x/2\rfloor+\lfloor x/3\rfloor+\cdots+\lfloor x/10000\rfloor$. Find $Q(2023)-Q(2022)$. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
A $5\times5$ square board has one identical-looking coin on each cell. Exactly two coins are counterfeit, of equal weight; all genuine coins have the same weight and are heavier than the counterfeit coins. The counterfeit coins occupy two cells sharing exactly one vertex. Can one guarantee identifying (a) 13 genuine coins; (b) 15 genuine coins; (c) 17 genuine coins, using only one weighing on a balance without weights? Original weights: (a) 2 points, (b) 3 points, (c) 2 points.
