Search found 230 matches
- Thu May 13, 2021 5:21 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks"
- Replies: 4
- Views: 6666
Re: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks"
Let $a_1 \leq a_2 \leq a_3 \cdots \leq a_n$ be a sequence of positive integers. For $1 \leq i \leq a_n$, let $b_i$ be the number of terms in the sequence that are not smaller than $i$. It is given that, $b_1 > b_2 > \cdots > b_{a_n}$, for each $1 \leq i \leq a_n, b_i$ is a power of three, and $b_1 ...
- Thu May 13, 2021 1:18 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks"
- Replies: 4
- Views: 6666
Re: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks"
Let $a_1 \leq a_2 \leq a_3 \cdots \leq a_n$ be a sequence of positive integers. For $1 \leq i \leq a_n$, let $b_i$ be the number of terms in the sequence that are not smaller than $i$. It is given that, $b_1 > b_2 > \cdots > b_{a_n}$, for each $1 \leq i \leq a_n, b_i$ is a power of three, and $b_1 ...
- Fri May 07, 2021 7:41 pm
- Forum: Algebra
- Topic: A question about FE
- Replies: 9
- Views: 15378
- Thu May 06, 2021 2:05 pm
- Forum: Algebra
- Topic: A question about FE
- Replies: 9
- Views: 15378
Re: A question about FE
Doesn't $x=0 \Rightarrow f(0)=0$? then the domain $\mathbb{R} \backslash \{0\}$ shouldn't mess around. :| SO I guess your function is $f(x)=\frac{1}{x}$ if $x \neq 0$, and $f(0)=0$ if $x=0$. I guess then it works. Actually the main question is does $f(xf(x))=xf(x) \Rightarrow f(x)=x\ \ \forall x \i...
- Thu May 06, 2021 4:02 am
- Forum: Algebra
- Topic: A question about FE
- Replies: 9
- Views: 15378
Re: A question about FE
Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk ) I think the problem with $f(x)=\frac{1}{x}$ is ...
- Wed May 05, 2021 9:09 pm
- Forum: Algebra
- Topic: A question about FE
- Replies: 9
- Views: 15378
Re: A question about FE
Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk ) I think the problem with $f(x)=\frac{1}{x}$ is ...
- Tue May 04, 2021 10:11 pm
- Forum: National Math Camp
- Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
- Replies: 3
- Views: 6208
- Mon May 03, 2021 9:18 pm
- Forum: National Math Camp
- Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
- Replies: 3
- Views: 6208
Re: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
We place some checkers on an $n\times n$ checkerboard so that they follow the conditions : Every square that does not contain a checker shares a side with one that does; Given any pair of squares that contain checkers, we can find a sequence of squares occupied by checkers that start and end with t...
- Mon May 03, 2021 8:30 pm
- Forum: National Math Camp
- Topic: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
- Replies: 2
- Views: 5847
Re: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
After seeing the solution, I was like "Keno parlam na!!!"
- Mon May 03, 2021 8:29 pm
- Forum: National Math Camp
- Topic: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
- Replies: 2
- Views: 5847
Re: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
I couldn't solve it myself :oops: . But since no one is posting the answer I am doing it. This problem came in the Putnam-2002(A3) Note that each of the sets $\{ 1\} ,\{ 2\} ,...,\{ n\}$ has the desired property. Moreover, for each set $S$ with integer average m that does not contain m, $S \cup {m}$...