AMO-Bench 2025 Problem 40

Answer

Given positive integers $n$ and $t$ with $1 \leq t < n$ . There are $n$ cups, all initially upright. An operation consists of choosing $t$ cups and turning them over. If it is possible to turn all $n$ cups upside down in $m$ operations, find the minimum value of $m$ . (expressed in terms of $n$ and $t$ )

Dataset text AMO-Bench. Open licence.

Show the answer

The answer can be expressed in various forms. Based on the values of $t$ , $n$ , and $m$ listed below, verify whether the student's answer is correct: \begin{enumerate} \item t = 500, n = 1000, then m = 2 \item t = 2, n = 100, then m = 50 \item t = 8, n = 10, then m = 3 \item t = 500, n = 999, then m has no solution \item t = 2, n = 99, then m has no solution \item t = 8, n = 9, then m has no solution \item t = 499, n = 1000, then m = 4 \item t = 3, n = 66, then m = 22 \item t = 15, n = 20, then m = 4 \item t = 97, n = 999, then m = 11 \item t = 33, n = 297, then m = 9 \item t = 3, n = 9, then m = 3 \end{enumerate}

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.

Add a solution

Solutions

Similar problems