Search found 312 matches
- Sun Dec 13, 2020 11:39 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 642385
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, pro...
- Thu Dec 10, 2020 10:28 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 642385
Re: FE Marathon!
$\ f(x)^2 = f(x) \times f(x)$
- Thu Dec 10, 2020 10:24 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 642385
Re: FE Marathon!
আমি যতদূর সম্ভব সার্চ দিয়ে কোনো FE ম্যারাথন খুঁজে পাই নি। তাই এই ম্যারাথন চালু করতেসি।সবাই এগিয়ে আসলে হয়তো এটা চালু হবে। হ্যাপি প্রব্লেম সলভিং! Problem 1: Find all functions $\ f$ such that $\ f: \mathbb{R} \rightarrow \mathbb{R}$ and $\ f(xf(x)+f(y))=f(x)^2+y $ for all real $x,\ y$. Good initia...
- Thu Oct 03, 2019 11:44 pm
- Forum: Geometry
- Topic: IGO official website
- Replies: 2
- Views: 45561
Re: IGO official website
https://igo-official.irMubin Hasan wrote: ↑Mon Sep 16, 2019 5:53 amDo you happen to know how can i get the resources and the past papers in english???
- Thu Jul 18, 2019 11:13 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P6
- Replies: 0
- Views: 42765
IMO 2019/P6
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of ...
- Thu Jul 18, 2019 11:11 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P5
- Replies: 0
- Views: 42285
IMO 2019/P5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, ...
- Thu Jul 18, 2019 11:05 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P4
- Replies: 1
- Views: 11249
IMO 2019/P4
Find all pairs $(k,n)$ of positive integers such that $k!=(2^n-1)(2^n-2)\cdots(2^n-2^{n-1})$
Proposed by Gabriel Chicas Reyes, El Salvador
Proposed by Gabriel Chicas Reyes, El Salvador
- Thu Jul 18, 2019 11:02 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P3
- Replies: 5
- Views: 67812
IMO 2019/P3
A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but ...
- Thu Jul 18, 2019 10:59 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P2
- Replies: 0
- Views: 42585
IMO 2019/P2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\...
- Thu Jul 18, 2019 10:57 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P1
- Replies: 1
- Views: 11033
IMO 2019/P1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$,
$f(2a)+2f(b)=f(f(a+b)).$
Proposed by Liam Baker, South Africa
$f(2a)+2f(b)=f(f(a+b)).$
Proposed by Liam Baker, South Africa