17

Part of 2022 BMT

Problems(2)

BMT 2022 Guts #17

Source:

Compute the number of ordered triples

(a, b, c)

of integers between

-100

100

inclusive satisfying the simultaneous equations

a^3 - 2a = abc - b - c

b^3 - 2b = bca - c - a

c^3 - 2c = cab - a - b.

algebrasystem of equationsnumber theory

BMT 2022 General p17

Source:

Midori and Momoi are arguing over chores. Each of

5

chores may either be done by Midori, done by Momoi, or put off for tomorrow. Today, each of them must complete at least one chore, and more than half of the chores must be completed. How many ways can they assign chores for today? (The order in which chores are completed does not matter.)