Search found 230 matches
- Mon Dec 21, 2020 11:54 pm
- Forum: Number Theory
- Topic: A problem from USAMO 2003
- Replies: 5
- Views: 7439
Re: A problem from USAMO 2003
Prove that for all positive integers n there is an n-digit multiple of 5^n all the the digits of which is odd. We will prove this by induction. But first, notice that the set of odd digits <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:5.339em"><span style=...
- Sun Dec 20, 2020 3:28 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
Find all functions $f$ $:$ $\Bbb Q \to \Bbb Q$ that satisfies,
\[f(x+y)=f(x)+f(y)+xy\]
\[f(x+y)=f(x)+f(y)+xy\]
- Fri Dec 18, 2020 10:32 pm
- Forum: Geometry
- Topic: 7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1
- Replies: 3
- Views: 7337
Re: 7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1
It took 1 hour at the exam hall and 25 minutes to recall it about a decade later :) Let us draw the bisectors of $\angle MND $ and $\angle MNC$ which intersects AB at X and Y respectively. we get the followings: $\angle ADN=\angle YNC= \angle MNY =\angle MYN$[Using AB||CD(YN transversal) and what is...
- Fri Dec 18, 2020 9:37 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
I am giving up :( :( :( :( :( I tried a lot for the last 2 days but couldn't solve it :( :( :( My progress: Let us denote the functional equation with (1) Let $x=0 $in (1) Then, $f(0)=f(yf(0)) \cdots (2) $ let $f(0) \neq 0$ then $f(0)=c$ for some real constant c Using (2) we can change y as whatever...
- Tue Dec 15, 2020 10:50 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2020 P11
- Replies: 3
- Views: 3967
Re: BdMO National Secondary 2020 P11
How did the idea of using Binary came? (The computer?) If someone tried to see what happens when we use binary he will get that binary forms a quite simple procedure and easy to notice. But the idea of trying with binary is not something I do in every problem. Do you try to use binary in every probl...
- Tue Dec 15, 2020 9:32 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
[N.B I am not sure the solution is correct. If somebody would kindly check it, I will be grateful.] Given that, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x<...
- Fri Dec 11, 2020 11:26 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
[N.B I am not sure the solution is correct. If somebody would kindly check it, I will be grateful.] Given that, $f(xf(x)+f(y))=f(x)^2+y \cdots (1)$ Let, $x = 0$ then, $f(0f(0)+f(y))=f(0)^2+y$ Or, $f(f(y))=f(0)^2+y$ Or, $f(f(y))=k^2+y [\text{Let, f(0)=k}] $ Now proving $f^2(y)$ is onto function, proo...
- Fri Dec 11, 2020 9:06 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
If we can prove $f(f(x))$ is onto can we prove $f(x)$ is onto too? If we can than how?
- Thu Dec 10, 2020 9:49 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 1408131
Re: FE Marathon!
Is $f(x)^2=f(x) \times f(x)$?
or $ f(x)^2=f(f(x))$?
or $ f(x)^2=f(f(x))$?
- Mon Dec 07, 2020 7:07 pm
- Forum: National Math Camp
- Topic: National Math Camp 2020 Exam 1 Problem 1
- Replies: 4
- Views: 8191
Re: National Math Camp 2020 Exam 1 Problem 1
We know, $a^3 + b^3+c^3-3abc=\frac{1}{2}(a+b+c)\{(a-b)^2+(b-c)^2+(c-a)^2\}$ $\therefore m^3+n^3+k^3-3mnk = \frac{1}{2}(m+n+k)\{(m-n)^2+(n-k)^2+(k-m)^2\}$ Or,$m^3+n^3+k^3-3mnk = \frac{1}{2}(m+n+k)\{mnk\}$ Or, $2m^3+2n^3+2k^3-6mnk= (m+n+k)(mnk)$ Or,$2(m^3+n^3+k^3)=(m+n+k)(mnk)+6mnk$ $\therefore 2m^3+2...