MathDB

6

Part of 2022 Princeton University Math Competition

Problems(1)

2022 PUMaC Team #6

Source:

9/9/2023
A sequence of integers x1,x2,...x_1, x_2, ... is double-dipped if xn+2=axn+1+bxnx_{n+2} = ax_{n+1} + bx_n for all n≥1n \ge 1 and some fixed integers a,ba, b. Ri begins to form a sequence by randomly picking three integers from the set {1,2,...,12}\{1, 2, ..., 12\}, with replacement. It is known that if Ri adds a term by picking anotherelement at random from {1,2,...,12}\{1, 2, ..., 12\}, there is at least a 13\frac13 chance that his resulting four-term sequence forms the beginning of a double-dipped sequence. Given this, how many distinct three-term sequences could Ri have picked to begin with?
algebracombinatorics