MathDB

23

Part of 2021 BMT

Problems(2)

BMT 2021 Guts Round p23

Source:

10/7/2022
Alireza is currently standing at the point (0,0)(0, 0) in the x−yx-y plane. At any given time, Alireza can move from the point (x,y)(x, y) to the point (x+1,y)(x + 1, y) or the point (x,y+1)(x, y + 1). However, he cannot move to any point of the form (x,y)(x, y) where y≡2x  (mod    5)y \equiv 2x\,\, (\mod \,\,5). Let pkp_k be the number of paths Alireza can take starting from the point (0,0)(0, 0) to the point (k+1,2k+1)(k + 1, 2k + 1). Evaluate the sum ∑k=1∞pk5k.\sum^{\infty}_{k=1} \frac{p_k}{5^k}..
combinatorics
BMT 2021 General p23

Source:

9/27/2023
Shivani has a single square with vertices labeled ABCDABCD. She is able to perform the following transformations: ∙\bullet She does nothing to the square. ∙\bullet She rotates the square by 9090, 180180, or 270270 degrees. ∙\bullet She reflects the square over one of its four lines of symmetry. For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
geometrycombinatorics