## Search found 244 matches

Wed Oct 17, 2012 9:48 pm
Topic: IMO MOCK5 (iii)
Replies: 7
Views: 3394

### Re: IMO MOCK5 (iii)

Phlembac Adib Hasan wrote:My solution is same as Najif, so there is no need to post it.
Same here Sat Oct 13, 2012 7:14 pm
Topic: IMO MOCK5 (iii)
Replies: 7
Views: 3394

### Re: IMO MOCK5 (iii)

Solution
Sat Oct 13, 2012 7:03 pm
Topic: IMO MOCK5 (iii)
Replies: 7
Views: 3394

### IMO MOCK5 (iii)

Let $O$ and $H$ be the circumcenter and orthocenter of acute $△ABC$. The bisector of $\angle BAC$ meets the circumcircle $Γ$ of $△ABC$ at $D$. Let $E$ be the mirror image of $D$ with respect to line $BC$. Let $F$ be on $Γ$ such that $DF$ is a diameter. Let lines $AE$ and $FH$ meet at $G$. Let $M$ be...
Sat Oct 13, 2012 6:54 pm
Topic: IMO MOCK5 (V)
Replies: 1
Views: 1711

My solution : MOCK 5 (v).JPG Let the circumcircle of the triangle $ABC$ intersect the line $MF$ at the point $F'$ (where $F',F$ lie on the same side of $BC$) and $BC \cap AF'=P'$ Now as $M$ is the midpoint of $BC$ and $MF'? BC$ , $\angle BF'M=\angle CF'M=\frac{1}{2} \angle BF'C=\frac{A}{2}$ So $\ang... Sat Oct 13, 2012 6:49 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO MOCK5 (V) Replies: 1 Views: 1711 ### IMO MOCK5 (V) In acute$△ABC$,$AB>AC$. Let$M$be the midpoint of$BC$. The exterior angle bisector of$\angle BAC$meets ray$BC$at$P$. Points$K$and$F$lie on line$PA$such that$MF⊥BC$and$MK⊥PA$. Prove that$BC^2=4PF.AK$Fri Oct 12, 2012 2:21 pm Forum: Secondary Level Topic:$3^{x}+4^{y}=5^{z}$Replies: 1 Views: 1354 ### Re:$3^{x}+4^{y}=5^{z}$Case 1 :$z$is non negative Then$x,y$will also be non negative Sub Case 1 :$x=0$Then$4^y+1=5^z$. As we know$v_5(4^y+1)=1+v_5(y)$...$(i)$let$y=5^k.a$Using$(i)4^{5^k.a}+1=5^{k+1}$that imply the only possible value of$k$is$0$, and$a=1$So Solution for this case is$(x,y,z)=(1,0,...
Mon Sep 24, 2012 5:30 pm
Forum: Algebra
Topic: Functional Equation
Replies: 1
Views: 1684

### Functional Equation

Find all $f:R \to R$ so that
$f(x^3+y^3)=xf(x^2)+yf(y^2)$ for all $x,y \in R$
Mon Sep 24, 2012 4:28 pm
Forum: Algebra
Topic: How's that FE?
Replies: 9
Views: 3141