Search found 665 matches
- Tue Feb 26, 2013 6:59 pm
- Forum: Geometry
- Topic: Prove concyclic
- Replies: 4
- Views: 4190
Prove concyclic
Let $ABCD$ be a cyclic quadrilateral. Let $AD \cap BC=E,AC \cap BD=F,EF \cap CD=G$. Let $M$ be the midpoint of $CD$. Prove that $A,B,M,G$ are con-cyclic.
- Sun Feb 24, 2013 7:24 pm
- Forum: Number Theory
- Topic: Find all integers x,y,z
- Replies: 1
- Views: 1925
Re: Find all integers x,y,z
Note that if $(x,y,z)$ is a solution then $(y,x,z),(-x,-y,z),(-y,-x,z),(x,-y,z),(-x,y,z)$ are also possible solutions. So WLOG let $x,y \in \mathbb{N} y \ge x$. $x^2+y^2+1=xyz \Rightarrow y^2-zxy+x^2+1=0$ which is a quadratic equation with variable $y$. Now using Vieta jumping we will get if $(y,x)$...
- Sun Feb 24, 2013 7:07 pm
- Forum: Geometry
- Topic: A Very Nice Problem
- Replies: 11
- Views: 8016
Re: A Very Nice Problem
Similar to APMO-2012-4. Here's the sketch of my solution. Lemma: In $\triangle ABC, P \in BC$. Then $\frac{MB}{MC}=\frac{AB}{AC}.\frac{\sin BAP}{\sin CAP}$. Use this lemma and sine law to prove $ABSC$ is a harmonic quadrilateral i.e. $\frac{AB}{AC}=\frac{BS}{SC}$. Use them again to prove $\frac{BS}{...
- Wed Feb 06, 2013 7:32 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
Problem 30: In triangle $ABC$, $w$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $w$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Pr...
- Tue Feb 05, 2013 5:11 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
Here's the correct problem statement: $Problem 29$: Fixed points $B$ and $C$ are on a fixed circle $w$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $w$ again in $K$. Tangent ...
- Sun Feb 03, 2013 10:29 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
Problem $\boxed {28}$: Trapezoid $ABCD$, with $AB$ parallel to $CD$, is inscribed in circle $w$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $w$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to $AB$ intersect $BD$ and $BC$ at points $R$ and $S$, r...
- Sun Feb 03, 2013 8:55 pm
- Forum: Higher Secondary Level
- Topic: Secondary and Higher Secondary Marathon
- Replies: 128
- Views: 296484
Re: Secondary and Higher Secondary Marathon
Probelm $37$: $\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$. Source: Iran NMO-2012-4. Note: In BdMO Summer Camp-2012, a similar problem was given in the Number Theory problem set. Zubaer vai...
- Sun Feb 03, 2013 8:53 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
$Problem 29$: Fixed points $B$ and $C$ are on a fixed circle $w$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $w$ again in $K$. Tangent in $A$ to circumcircle of triangle $A...
- Sun Feb 03, 2013 7:46 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
Problem $\boxed {28}$: Trapezoid $ABCD$, with $AB$ parallel to $CD$, is inscribed in circle $w$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $w$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to $AB$ intersect $BD$ and $BC$ at points $R$ and $S$, r...
- Sat Feb 02, 2013 9:14 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 113362
Re: IMO Marathon
$P(0,c) \Rightarrow f(c)=f(0)^2+\alpha.c \Rightarrow c=\frac{-f(0)^2}{\alpha}$.....$(1)$ $P(x,\frac{-f(x^2)}{\alpha}) \Rightarrow f(x^2-\frac{f(x)^2}{\alpha)}+f(\frac{-f(x)^2}{\alpha}))=0$.....$(2)$ Let $f(c_1)=f(c_2)=0$ $P(0,c_1),P(0,c_2)$ implies $c_1=c_2$, so from $(1),(2)$ we conclude $f(\frac{...