BdMO Regional 2021 Higher Secondary P6
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Let $S=\{1,2,3,\dots,15\}$, How many sets $X\subseteq S$ are there such that if $x\in X$ and $3x\in S$, then $3x\in X$?
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Re: BdMO Regional 2021 Higher Secondary P6
The problem statement created confusion among the students, so here's a detailed clarification :
Solution :
We have to focus on these subsets of $S$ :
Now we have to choose elements from $A,B,C,D,E$ such that they create the set $X$.
We have to focus on these subsets of $S$ :
- $A=\{1,3,9\}$
- $B=\{2,6\}$
- $C=\{4,12\}$
- $D=\{5,15\}$
- $E=\{7,8,10,11,13,14\}$
Now we have to choose elements from $A,B,C,D,E$ such that they create the set $X$.
- From $A$, we can either choose $1,3,9$ or only $3,9$ or only $9$ or none. (Total $4$ choices)
- From $B$, we can either choose $2,6$ or only $6$ or none. (Total $3$ choices)
- From $C$, we can either choose $4,12$ or only $12$ or none. (Total $3$ choices)
- From $D$, we can either choose $5,15$ or only $15$ or none. (Total $3$ choices)
- From $E$, we can choose any element we want and there are total $2^6$ choices for that.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann