11

Part of 2024 AIME

Problems(2)

Source: 2024 AIME I #11

Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is

\tfrac{m}{n}

m

n

are relatively prime positive integers. What is

m+n

Nerfed 2023 JMO 1

Source: 2024 AIME II P11

Find the number of triples of nonnegative integers

(a,b,c)

a + b + c = 300

a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.

AIMEAIME IAIME IIHi