Problem

2026 AIME I Problem 15

Let a, b, and n be positive integers with both a and b greater than or equal to 2 and less than or equal to 2n.{} Define an a \times b cell loop in a 2n \times 2n grid of cells to be the 2a + 2b - 4 cells that surround an (a - 2) \times (b - 2) (possibly empty) rectangle of cells in the grid. For example, the following diagram shows a way to partition a 6 \times 6 grid of cells into 4 cell loops. [asy] usepackage(“mathptmx”); size(6cm); dotfactor = 30; fill((0, 2)--(0, 6)--(4, 6)--(4, 2)--cycle, royalblue); fill((1, 3)--(1, 5)--(3, 5)--(3, 3)--cycle, lightred); fill((4, 0)--(4, 6)--(6, 6)--(6, 0)--cycle, lightyellow); draw((1, 1)--(3, 1)^^(5, 1)--(5, 5)^^(1, 3)--(1, 5)--(3, 5)--(3, 3)--cycle^^(0, 0)--(0, 6)--(6, 6)--(6, 0)--cycle^^(0, 2)--(4, 2)^^(4, 0)--(4, 6), linewidth(4)); dot((2, 4)); for (int i = 1; i <= 5; i += 1) { draw((0, i)--(6, i)^^(i, 0)--(i, 6)); } [/asy] Find the number of ways to partition a {}10 \times 10 grid of cells into 5 cell loops so that every cell of the grid belongs to exactly one cell loop.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.


Show/Hide Problem Tags

Problem Tags: Counting and probability
Want to contribute problems and receive full credit? Click here to add your problem!
Please report any issues to us in our Discord server
Go to previous contest problem (SHIFT + Left Arrow)
Category: AIME I
Points: 6
Back to practice
Train more like this