3

Part of 2018 Online Math Open Problems

Problems(2)

2017-2018 Spring OMO Problem 3

Source:

Hen Hao randomly selects two distinct squares on a standard

8\times 8

chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form

\frac mn

for relatively prime positive integers

m

n

100m+n

.
Proposed by James Lin

2018-2019 Fall OMO Problem 3

Source:

Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
Proposed by Michael Ren