2

Part of 2021 MMATHS

Problems(2)

MMATHS 2021, Problem 2: Digital Reductions

Source:

Define the digital reduction of a two-digit positive integer

\underline{AB}

to be the quantity

\underline{AB} - A - B

. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of

62

62 - 6 - 2 = 54.

)
Proposed by Andrew Wu

MMATHS 2021, TB Problem 2: Expected Value of a Grid

Source:

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's score is the sum of all numbers recorded this way. Deyuan shades each square in a blank

n\times n

grid with probability

k

; he notices that the expected value of the score of the resulting grid is equal to

k

, too! Given that

k > 0.9999

, find the minimum possible value of

n

.
Proposed by Andrew Wu