HMMT 十一月 2025 · 团队赛 · 第 11 题

HMMT November 2025 November 08, 2025 Team Round

1. [20] Compute the number of ways to divide an 8 × 8 square into 3 rectangles, each with (positive) integer side lengths.
2. [25] Mark writes the squares of several distinct positive integers (in base 10) on a blackboard. Given that each nonzero digit appears exactly once on the blackboard, compute the smallest possible sum of the numbers on the blackboard.
3. [30] Let ABCD and CEF G be squares such that C lies on segment DG and E lies on segment BC . Let O be the circumcenter of triangle AEG . Given that A , D , and O are collinear and AB = 1, compute F G .
4. [35] For positive integers n and k with k > 1, let s ( n ) denote the sum of the digits of n when written k in base k . (For instance, s (2025) = 5 because 2025 = 2210000 .) A positive integer n is a digiroot if 3 3 p s ( n ) = s ( n ). Compute the sum of all digiroots less than 1000. 2 4
5. [40] Kelvin the frog is in the bottom-left cell of a 6 × 6 grid, and he wants to reach the top-right cell. He can take steps either up one cell or right one cell. However, there is a raccoon in one of the 36 cells uniformly at random, and Kelvin’s path must avoid this raccoon. Compute the expected number of distinct paths Kelvin can take to reach the top-right cell. (If the raccoon is in either the bottom-left or top-right cell, then there are 0 such paths.) ◦
6. [45] Let P be a point inside triangle ABC such that BP = P C and ∠ ABP + ∠ ACP = 90 . Given that AB = 12, AC = 16, and AP = 11, compute the area of the concave quadrilateral ABP C .
7. [45] Let S be the set of all positive integers less than 143 that are relatively prime to 143. Compute the number of ordered triples ( a, b, c ) of elements of S such that a + b = c .
8. [50] Alexandrimitrov is walking in the 3 × 10 grid below. He can walk from a cell to any cell that shares an edge with it. Given that he starts in cell A , compute the number of ways he can walk to cell B such that he visits every cell exactly once. (Starting in cell A counts as visiting cell A .) A B
9. [55] Let a , b , and c be positive real numbers such that √ √ √ a + b + c = 7 , √ √ √ a + 1 + b + 1 + c + 1 = 8 , √ √ √ √ √ √ ( a + 1 + a )( b + 1 + b )( c + 1 + c ) = 60 . Compute a + b + c .
10. [55] Let ABCD be an isosceles trapezoid with AB parallel to CD , and let P be a point in the interior of ABCD such that ∠ P BA = 3 ∠ P AB and ∠ P CD = 3 ∠ P DC. 2 Given that BP = 8, CP = 9, and cos ∠ AP D = , compute cos ∠ P AB . 3 © 2025 HMMT