Let $P(x,y)$ denote the FE. Note that if $f$ is a solution then $-f$ is also a solution. Now, $P(2,2)$ imply that $f(f(2)^2)=0$. Let $t=f(2)^2$. Case 1: $t \neq 1$ $P(x,t)$ implies $f(0)+f(x+t)=f(xt)$, and we can find $x$ so that $x+t=xt$. So $f(0)=0$. Now $P(x,0)$ implies $f(x)=0$ for all real $x$....

- Wed Jul 19, 2017 1:17 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2017 Problem 2
- Replies:
**3** - Views:
**583**

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y:$

\[f(f(x)f(y)) + f(x+y) = f(xy).\]

\[f(f(x)f(y)) + f(x+y) = f(xy).\]

- Wed Jul 19, 2017 12:36 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2017 Problem 2
- Replies:
**3** - Views:
**583**

Zawadx wrote:*** Bwwaaahhh infinities

But $f(n)|f(0)$ and $f(0)>0$, doesn't that mean $f$ does achieve a maximal value? So your solution is correct I think.

- Wed Apr 26, 2017 7:39 pm
- Forum: Number Theory
- Topic: IMO Shortlist 2011 N5
- Replies:
**4** - Views:
**341**

Zawadx wrote:There's a typo in the determinant: zero for you~

Edited. Latexing a determinant is a pain in the first place, locating these typos are difficult

It was correct in my paper, so if I had submitted, it wouldn't have been a zero, rather a seven.

- Mon Apr 24, 2017 1:45 pm
- Forum: Geometry
- Topic: USA(J)MO 2017 #3
- Replies:
**6** - Views:
**315**

We use barycentric coordinates. Let $P\equiv (p:q:r)$. Now, we know that $pq+qr+rp=0$ [The equation of circumcircle for equilateral triangles]. Now, $D\equiv (0:q:r), E\equiv (p:0:r), F\equiv (p:q:0)$. So, the area of $\triangle DEF$ divided by the area of $\triangle ABC$ is: $$\dfrac{1}{(p+q)(q+r)(...

- Sat Apr 22, 2017 6:23 pm
- Forum: Geometry
- Topic: USA(J)MO 2017 #3
- Replies:
**6** - Views:
**315**

aritra barua wrote:When $a,b$ € $N$ and it follows that $a^x$=$b^y$,there exists $t$ € $N$ such that $a$=$t^k$,$b$=$t^q$.This lemma can be quite handy in this problem.

Are you sure that helps? $a$ and $b$ need to be coprime.

- Sat Apr 22, 2017 3:19 pm
- Forum: Number Theory
- Topic: USAJMO/USAMO 2017 P1
- Replies:
**3** - Views:
**232**

For calculating the tangent, See that the derivative of the function at the point $2$ is $-\dfrac{1}{12}$, and plugging $2$ gives us $\dfrac{1}{12}$. So, the equation of the tangent is $-\dfrac{x}{12}+c$, and it goes through $(2,\dfrac{1}{12})$. This lets us determine the equation.

- Sat Apr 22, 2017 11:24 am
- Forum: Algebra
- Topic: At last an ineq USAMO '17 #6
- Replies:
**3** - Views:
**180**

Here we use the well known tangent line trick. It is easy to see that the minimum is achieved at $(a,b,c,d)=(0,0,2,2)$ and its cyclic variants. Also, the most troublesome thing in this inequality is the quantities of the form $\dfrac{1}{x^3+4}$. So, we draw a tangent of $\dfrac{1}{x^3+4}$ at the poi...

- Sat Apr 22, 2017 11:19 am
- Forum: Algebra
- Topic: At last an ineq USAMO '17 #6
- Replies:
**3** - Views:
**180**

If there is a tie at the end of the day phase, the votes are resolved in this order (The person who was voted on by the earliest role in this list is lynched): Bashbaba > Determinant Detector > darij grinberg > Telv Cohl > Gonju > Geo Hater > Projective Priest > Angle Chaser > Figure Scout

- Thu Apr 20, 2017 9:16 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**27** - Views:
**1258**

I propose that everyone except gonju votes himself.

We should take advantage of the voting ordering.

We should take advantage of the voting ordering.

- Thu Apr 20, 2017 9:14 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**27** - Views:
**1258**