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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Sequences and Series

Olympiad workbook · IOQM · Sequences and Series

Starting with a positive integer M written on the board, Alice...

Starting with a positive integer

M

written on the board, Alice plays the following game: in each move, if

x

is the number on the board, she replaces it with

3 x+2

. Similarly, starting with a positive integer

N

written on the board, Bob plays the following game: in each move, if

x

is the number on the board, he replaces it with

2 x+27

. Given that Alice and Bob reach the same number after playing 4 moves each, find the smallest value of

M+N

.

Correct answer

10

Step-by-step solution

In 4 steps Alice will write

(3(3(3(3 M+2)+2)+2)+2)=\alpha

and Bob will write

(2(2(2(2 N+27)+27)+27)+27)=\beta

As

\alpha=\beta

, we get

\begin{array}{ll} & 81 M=16 N+325 \\ \therefore \quad & M_{\min }=5 \text { and } N_{\min }=5 \\ & (M+N)_{\min }=10 \end{array}

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