MathDB

Tie 3

Part of 2022 BMT

Problems(4)

BMT 2022 Algebra Tiebreaker p3

Source:

9/27/2023
Tej writes 2,3,...,1012, 3, ..., 101 on a chalkboard. Every minute he erases two numbers from the board, xx and yy, and writes xy/(x+y−1)xy/(x+y-1). If Tej does this for 9999 minutes until only one number remains, what is its maximum possible value?
algebra
BMT 2022 Discrete Tiebreaker p3

Source:

9/27/2023
Let AA be the product of all positive integers less than 10001000 whose ones or hundreds digit is 77. Compute the remainder when A/101A/101 is divided by 101101.
combinatoricsnumber theory
BMT 2022 Geometry Tiebreaker #3

Source:

8/12/2023
In triangle △ABC\vartriangle ABC, MM is the midpoint of AB‾\overline{AB} and NN is the midpoint of AC‾\overline{AC}. Let PP be the midpoint of BN‾\overline{BN} and let QQ be the midpoint of CM‾\overline{CM}. If AM=6AM = 6, BC=8BC = 8 and BN=7BN = 7, compute the perimeter of triangle △NPQ\vartriangle NP Q.
geometry
BMT 2022 General Tiebreaker #3

Source:

3/9/2024
You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices, but is not adjacent to any other edges or vertices. Each edge is adjacent to both of its vertices, but is not adjacent to any other vertices. What is the minimum number of colors required for a coloring satisfying this property?
combinatorics