Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that

\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]for all real numbers $x$ and $y$.

## Search found 1004 matches

- Thu Aug 15, 2019 8:46 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2019 P5
- Replies:
**0** - Views:
**329**

- Thu Aug 15, 2019 8:45 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2019 P4
- Replies:
**0** - Views:
**232**

### APMO 2019 P4

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculatio...

- Thu Aug 15, 2019 8:44 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2019 P3
- Replies:
**0** - Views:
**237**

### APMO 2019 P3

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a se...

- Thu Aug 15, 2019 8:42 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2019 P2
- Replies:
**0** - Views:
**240**

### APMO 2019 P2

Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have $$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$For each $m$, det...

- Thu Aug 15, 2019 8:41 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2019 P1
- Replies:
**0** - Views:
**244**

### APMO 2019 P1

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

- Thu May 16, 2019 10:40 am
- Forum: Physics
- Topic: BdPhO Regional (Dhaka-South) Higher Secondary 2019/2
- Replies:
**3** - Views:
**850**

### Re: BdPhO Regional (Dhaka-South) Higher Secondary 2019/2

How did you draw this?

- Thu May 16, 2019 10:37 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2018:Full Solution
- Replies:
**2** - Views:
**1044**

### Re: BdMO National Secondary 2018:Full Solution

From the facebook page of Parallel Math School.SINAN EXPERT wrote: ↑Thu Apr 25, 2019 8:19 pmHow did you know about that?samiul_samin wrote: ↑Mon Feb 25, 2019 1:45 pmAll solution of the problems of BdMO National Secondary $2018$ are available here.

Made by Soyeb Pervez Jim

I mean this file is posted by Soyeb Pervez Jim.

### Re: 0^0=1!

Why $64\times 0^0=64$?SINAN EXPERT wrote: ↑Wed Apr 24, 2019 4:57 pmYeah, that's not any joke! I'm really gonna prove that!

We know, $2^6=64$

Again, $2^6=(2+0)^6=6_{C_0}2^60^0+6_{C_1}2^50^1+6_{C_2}2^40^2+...=64*0^0$ $⇒64=64*0^0$

Which is a clear proof of $0^0=1$.

True to say, I haven't found any mistake here.

- Wed Apr 10, 2019 10:18 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2019/9
- Replies:
**2** - Views:
**643**

### Re: BdMO National Higher Secondary 2019/9

No need to draw a figure?

- Wed Apr 10, 2019 10:17 am
- Forum: Combinatorics
- Topic: how to solve this?
- Replies:
**2** - Views:
**698**