If $k\neq\pm1$ then $g$ is not surjective.fahim_faiaz_adib wrote: ↑Mon Jun 21, 2021 4:56 pmNow let define function g(x)=kx.
Clearly g is surjective.
Search found 264 matches
- Tue Jun 22, 2021 10:13 am
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31916
Re: Special Problem Marathon
- Thu Jun 17, 2021 11:42 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31916
Problem 13
Let $k$ be a fixed positive integer. Let $\mathbb{N}$ be the set of all positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the condition \[\sum_{i=1}^{k}f^i(n)=kn\] For all $n\in\mathbb{N}$. Here $f^i(n)$ is the $i$-th iteration of $f$. Prove that $\forall n\in\mathbb{N}, f(n)=n$.
- Thu Jun 17, 2021 10:53 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31916
Re: Problem 12
Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircl...
- Wed Jun 16, 2021 8:16 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
- Replies: 5
- Views: 8617
Re: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Let $ (a,N)=x , (b,N)=y$ then $ (x,y)=1$ By Dirichlet's theorem there exists infinitely many primes $p,q$ such that $ p \equiv \dfrac{a}{x}$ (mod $\dfrac{N}{x}$) and $q \equiv \dfrac{b}{y}$ (mod $\dfrac {N}{y}$) or, $px \equiv a $ (mod N) and $ qy \equiv b $ (mod N) Setting $ m=px ,n = qy$ we are d...
- Thu Jun 10, 2021 9:42 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P1
- Replies: 2
- Views: 8522
Re: APMO 2021 P1
Let's assume $\lfloor x\rfloor=n, n\in\mathbb{Z}$. So we have $x=\sqrt{nr}>0\Longleftrightarrow n>0\Longleftrightarrow n\geq1$. So, the given condition is equivalent to the inequality, \begin{equation} \begin{split} &n\leq\sqrt{nr}<n+1\\ \Longleftrightarrow &n\leq r<n+2+\frac1n \end{split} \end{equa...
- Wed Jun 09, 2021 11:01 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Junior 2020 P10
- Replies: 6
- Views: 14278
- Wed Jun 09, 2021 5:27 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P5 - Determine all such functions
- Replies: 0
- Views: 7533
APMO 2021 P5 - Determine all such functions
Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.
- Wed Jun 09, 2021 5:23 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P4 - There exists a good subset consisting of at least $666$ cells
- Replies: 0
- Views: 7432
APMO 2021 P4 - There exists a good subset consisting of at least $666$ cells
Given a $32\times32$ table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a su...
- Wed Jun 09, 2021 5:19 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P3 - Geometry
- Replies: 2
- Views: 4679
APMO 2021 P3 - Geometry
Let $ABCD$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $AC$ and $BD$, let $L$ be the center of the circle tangent to sides $AB, BC$ and $CD$, and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A$ and $D$. Prove ...
- Wed Jun 09, 2021 5:01 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P2 - Determine all such polynomials
- Replies: 0
- Views: 7239
APMO 2021 P2 - Determine all such polynomials
For a polynomial $P$ and a positive integer $n$, define $P_n$ as the number of positive integer pairs $(a, b)$ such that $a < b \leq n$ and $\vert P(a)\vert-\vert P(b)\vert$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that $P_n \leq 2021$ for all positive integer...