## Search found 64 matches

Wed Apr 11, 2018 10:52 pm
Forum: Algebra
Topic: FE FE FE
Replies: 1
Views: 3917

$f(x)f(y+k)=2f(x+(y+k)f(x))=2f(x+yf(x)+\frac{2k}{f(y)}f(x+yf(x))$ [Recall, $f(x)=\frac{2}{f(y)}f(x+yf(x)$] $\Rightarrow f(x)f(y+k) = f(x+yf(x))f(\frac{2k}{f(y)})=\frac{1}{2}f(x)f(y)f(\frac{2k}{f(y)})$ $\Rightarrow 2f(y+k)=f(y)f(\frac{2k}{f(y)})=2f(y+\frac{2k}{f(y)}f(y))=2f(y+2k)$ Inductivly, $f(y+k)... Wed Apr 11, 2018 10:51 pm Forum: Algebra Topic: FE FE FE Replies: 1 Views: 3917 ### FE FE FE We denote by$\mathbb{R}^+$the set of all positive real numbers. Find all functions$f: \mathbb R^ + \rightarrow\mathbb R^ +$which have the property: $f(x)f(y)=2f(x+yf(x))$ for all positive real numbers$x$and$y$. Mon Apr 09, 2018 9:33 am Forum: Geometry Topic: China National Olympiad 2018 P4 Replies: 2 Views: 7974 ### Re: China National Olympiad 2018 P4 China 2018 #4.png Let$X$and$Y$be the midpoints of shorter arcs$AD$and$BC$in the circles$\circ ADPE$and$\circ BCPF$respectively.$Z$is the intersection point of$XE$and$YF$. It is obvious that$X, P, Y$are collinear. Claim 1:$\triangle AXP$and$\triangle BYP$are similar. Proof:$\...
Mon Apr 09, 2018 9:31 am
Forum: Geometry
Topic: China National Olympiad 2018 P4
Replies: 2
Views: 7974

### China National Olympiad 2018 P4

$ABCD$ is a cyclic quadrilateral whose diagonals intersect at $P$. The circumcircle of $\triangle APD$ meets segment $AB$ at points $A$ and $E$. The circumcircle of $\triangle BPC$ meets segment $AB$ at points $B$ and $F$. Let $I$ and $J$ be the incenters of $\triangle ADE$ and $\triangle BCF$, resp...
Wed Apr 04, 2018 10:17 pm
Forum: Number Theory
Topic: A Conjecture
Replies: 1
Views: 3769

### Re: A Conjecture

Doesn't work when $p=127$.
Tue Apr 03, 2018 12:26 pm
Forum: Geometry
Topic: What is the distance?
Replies: 3
Views: 8045

### Re: What is the distance?

I'm avoiding the calculations how I got that $PX=2R=65/2$ (By Heron's formula we got the area of the triangle $ABC$, then found out the height from on vertix to oppsite side, and then applied Brahmagupta's theorem.) Using the fact that, $CS/PS=XS/CS$ we get $CS=10$ I think you've made a typo here. ...
Fri Mar 30, 2018 10:22 pm
Topic: 2007 number 5 - divisibility
Replies: 1
Views: 5358

$4ab-1|(4a^2-1)^2 \Rightarrow 4ab-1|(a-b)^2$. Assume contradiction that $a \neq b$. wlog, $a>b$ $\frac{(a-b)^2}{4ab-1}=k \Rightarrow (a-b)^2=4abk-k$. Let $a$ and $b$ be such a solution such that $a+b$ is minimal. Consider the quadratic equation $$(x-b)^2=4xbk-k$$ $\Rightarrow x^2 - x(2b+4bk)+(b^2+k)... Sun Mar 25, 2018 9:29 pm Forum: Algebra Topic: FE from USAMO 2002 Replies: 4 Views: 5121 ### Re: FE from USAMO 2002 Let$P(x,y)$denotes the assertion.$P(0,0) \Rightarrow f(0)=0P(x,0) \Rightarrow f(x^2)=xf(x)$So,$f(a-b)=f(a)-f(b)$when$a,b \in \mathbb{R^+}$It can be proved that this statement is true for negatives too. That is,$f(a-b)=f(a)-f(b)$for all$a,b \in \mathbb{R}$Now,$P(x+1,x) \Rightarrow f(2...
Sun Mar 25, 2018 9:13 pm
Forum: Algebra
Topic: FE from USAMO 2002
Replies: 4
Views: 5121

### FE from USAMO 2002

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2 - y^2) = x f(x) - y f(y)$ for all pairs of real numbers $x$ and $y$
Thu Mar 15, 2018 2:50 pm
Forum: Algebra
Topic: Functional equation from Japan MO 2016
Replies: 1
Views: 3906

### Re: Functional equation from Japan MO 2016

Let $P(x,y)$ denotes the assertion. $P(0,0) \Rightarrow f(0) = (f(0))^2 \Rightarrow f(0) = 0$ or $1$ Case 1: $f(0)=0$ $P(x,0) \Rightarrow f(-x) = 2x \Rightarrow f(x) = -2x$ which is obviously a solution. Case 2: $f(0)=1$ $P(x,0) \Rightarrow f(-x) = f(x) + 2x$ ................... (i) \$P(x,-y) \Right...