Secondary and Higher Secondary Marathon
 Tahmid Hasan
 Posts: 665
 Joined: Thu Dec 09, 2010 5:34 pm
 Location: Khulna,Bangladesh.
Re: Secondary and Higher Secondary Marathon
Time for a new problem.
Problem $31$: Given a trapezoid $ABCD (AD \parallel BC)$ with $\angle ABC > 90^\circ$. Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
Source: International Zhautykov Olympiad 20131.
Problem $31$: Given a trapezoid $ABCD (AD \parallel BC)$ with $\angle ABC > 90^\circ$. Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
Source: International Zhautykov Olympiad 20131.
বড় ভালবাসি তোমায়,মা
 zadid xcalibured
 Posts: 217
 Joined: Thu Oct 27, 2011 11:04 am
 Location: mymensingh
Re: Secondary and Higher Secondary Marathon
Actually $\triangle{MO_{1}D}$ and $\triangle{MO_{2}C}$ are spirally symmetric.So are $\triangle{MO_{1}O_{2}}$ and
$\triangle {MDC}$.Now assuming the contrary we can show a contradiction.
$\triangle {MDC}$.Now assuming the contrary we can show a contradiction.
 zadid xcalibured
 Posts: 217
 Joined: Thu Oct 27, 2011 11:04 am
 Location: mymensingh
Re: Secondary and Higher Secondary Marathon
This solution process can be used as a lemma.
 zadid xcalibured
 Posts: 217
 Joined: Thu Oct 27, 2011 11:04 am
 Location: mymensingh
Re: Secondary and Higher Secondary Marathon
Problem $32$:Consider $\omega$ to be the circumcircle of $\triangle{ABC}$.$D$ is the midpoint of arc $BAC$ and $I$ is the incentre.Let $DI$ intersect $BC$ at $E$ and $\omega$ again at $F$.Let $P$ be a point on the line $AF$ such that $PE$ is parallel to $AI$.Prove that $PE$ is the bisector of $\angle{BPC}$.
Re: Secondary and Higher Secondary Marathon
I have solved it using angle chasing.Tahmid Hasan wrote:Time for a new problem.
Problem $31$: Given a trapezoid $ABCD (AD \parallel BC)$ with $\angle ABC > 90^\circ$. Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
Source: International Zhautykov Olympiad 20131.
$\angle O_{1}ND=\angle O_{1}MD=\angle O_{1}DM=\frac{1}{2}(180^{\circ}\angle MO_{1}D)=90^{\circ}\angle BAD$
$=180^{\circ}\angle BAD90^{\circ}=\angle ABC90^{\circ}=\angle MBC90^{\circ}=90^{\circ}\frac{1}{2}\angle MO_{2}C= \frac{1}{2}(180^{\circ}\angle MO_{2}C)=\angle O_{2}CM=\angle O_{2}MC$
$=\angle O_{2}NC$.
Also $\angle DNM+\angle CNM=180^{\circ}\angle MO_{1}D+MO_{2}C$
$=180^{\circ}2\angle BAD+360^{\circ}2\angle ABC=540^{\circ}2\times 180^{\circ}=180^{\circ}$ so $D,N,C$ are on a single line. Thus $O_{1},N,O_{2}$ will also be collinear.
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Re: Secondary and Higher Secondary Marathon
@Zadid vai,you haven't posted the source of Problem $\boxed {32}$
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 zadid xcalibured
 Posts: 217
 Joined: Thu Oct 27, 2011 11:04 am
 Location: mymensingh
Re: Secondary and Higher Secondary Marathon
Well,this is problem 3 of geomtry problem set of BDMC 2012.
 Tahmid Hasan
 Posts: 665
 Joined: Thu Dec 09, 2010 5:34 pm
 Location: Khulna,Bangladesh.
Re: Secondary and Higher Secondary Marathon
It's an Iran TST2012 problem given in the last camp pset. That time I made a really complicated solution using harmonic division with a little help from Fahim vai, later I made another one a lot simpler. Here it isSANZEED wrote:@Zadid vai,you haven't posted the source of Problem $\boxed {32}$
Let $D'$ be the midpoint of arc $BC$ not containing $A$. So $DD' \perp BC$.
Let $DD' \cap PE=Q$. So $\angle PFD=\angle AD'D=\angle PQD \Rightarrow PFQD$ is concyclic.
So $PE.EQ=FE.ED=BE.CE \Rightarrow PBQC$ is concyclic. Since $BQ=CQ, PE$ bisects $\angle BPC$.
Note: This problem holds for any point in segment $AD'$ not particularly $I$.
It's very late at night now, I don't have any good problem to post. Maybe tomorrow.
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 Nadim Ul Abrar
 Posts: 244
 Joined: Sat May 07, 2011 12:36 pm
 Location: B.A.R.D , kotbari , Comilla
Re: Secondary and Higher Secondary Marathon
@Tahmid . Love you bro .
Problem $\boxed {33}$
Let $ABC$ be a right triangle ($\angle C=90$),let $CD$ be the hight and let $K$ be a point in the triangle such that $AK=AC$.Prove that the diametar of the curcumcircle of triangle $ABK$ that contains point $A$ is perpendicular to $DK$
Source : Serbia 1990
Problem $\boxed {33}$
Let $ABC$ be a right triangle ($\angle C=90$),let $CD$ be the hight and let $K$ be a point in the triangle such that $AK=AC$.Prove that the diametar of the curcumcircle of triangle $ABK$ that contains point $A$ is perpendicular to $DK$
Source : Serbia 1990
$\frac{1}{0}$
 Phlembac Adib Hasan
 Posts: 1016
 Joined: Tue Nov 22, 2011 7:49 pm
 Location: 127.0.0.1
 Contact:
Re: Secondary and Higher Secondary Marathon
Can you please provide a figure?Nadim Ul Abrar wrote:@Tahmid . Love you bro .
Problem $\boxed {33}$
Let $ABC$ be a right triangle ($\angle C=90$),let $CD$ be the hight and let $K$ be a point in the triangle such that $AK=AC$.Prove that the diametar of the curcumcircle of triangle $ABK$ that contains point $A$ is perpendicular to $DK$
Source : Serbia 1990
And are you sure this statement is correct and you didn't make any typo?
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