Search found 176 matches
- Mon Feb 11, 2013 3:53 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2013 Higher Secondary Problem 9
- Replies: 3
- Views: 4284
BdMO 2013 Higher Secondary Problem 9
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AD, BE, CF$ is a diametre. Extended $AB$ and $DC$ meet at $Z$, $CD$ and $FE$ at $X$, $EF$ and $BA$ at $Y$. $AC$ and $BF$ meets at $P$, $CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R$. If $O$ is the point of intersection of $Y...
- Mon Feb 11, 2013 3:43 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Higher Secondary 8
- Replies: 6
- Views: 6234
BdMO 2013 Higher Secondary Problem 8
$ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$, and $CF$. The line parallel to $DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33^\circ$, find $\an...
- Wed Feb 06, 2013 6:02 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
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 $BC$. Pro...
- Tue Feb 05, 2013 9:14 pm
- Forum: Higher Secondary Level
- Topic: Secondary and Higher Secondary Marathon
- Replies: 128
- Views: 301052
Re: Secondary and Higher Secondary Marathon
Problem 38:
$3^x+7^y=n^2$
how many integer solutions for $(x,y)$ are there?
$3^x+7^y=n^2$
how many integer solutions for $(x,y)$ are there?
- Mon Feb 04, 2013 9:43 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
Re: IMO Marathon
Yeah, Tahmid, it's from Iran NMO. Well, as you've already done it, wait to post your solution. If no one posts the solution, then post yours.
- Sun Feb 03, 2013 9:45 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
Re: IMO Marathon
Really sorry for the typo. It'd be circumcircle of triangle $ABH$. Forgive me. :-/
- Sun Feb 03, 2013 5:10 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
Re: IMO Marathon
Very nice solution, Nadim. Mine's quite dirty, but uses same concepts. :) $\text{Problem }29$: 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 $DH$ i...
- Sat Feb 02, 2013 11:39 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
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 $S$, re...
- Sun Jan 20, 2013 6:18 pm
- Forum: Higher Secondary Level
- Topic: Secondary and Higher Secondary Marathon
- Replies: 128
- Views: 301052
Re: Secondary and Higher Secondary Marathon
In 2012 national camp, Sourav solved a slightly different version of this problem using some cool rotation! Mugdho vaia solved it using co-ordinates though.
- Mon Nov 12, 2012 2:27 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114679
Re: IMO Marathon
I am taking the liberty and posting a problem. :/ Problem 6: Point $D$ lies inside triangle $ABC$ such that $\angle DAC =\angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE\perp EF$. Source:...