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Re: MPMS Problem Solving Marathon

For problem 1,if we substitute $a^2$=$4k$ where $k$ is an integer,we can easily find that $p$ |$a^2$+$1$.
by aritra barua
Mon May 29, 2017 3:53 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 2
Views: 119

Re: APMO 2017 P2

Let the reflection of $DM$ over $M$ be $D'M$.Thus we concurr that $AD'BD$ is a parallelogram as $M$ is designed to be the midpoint of $AB$.Let the midpoint of $AC$ be $X$.Thus we have $AX$=$CX$; $ZX$ as a common side of triangles $ZXA$ and $ZXC$;$\angle ZXA$=$\angle ZXC$.So,by $SAS$,we have $\bigtri...
by aritra barua
Mon May 29, 2017 12:34 am
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P2
Replies: 1
Views: 45

JBMO:NT

Consider the equation:$2^x$.$3^y$-$5^w$.$7^z$=$1$;$x$, $y$, $z$, $w$ are non negative integers.
by aritra barua
Tue Apr 25, 2017 4:10 pm
 
Forum: Number Theory
Topic: JBMO:NT
Replies: 1
Views: 104

Re: USAJMO/USAMO 2017 P1

When $a,b$ € $N$ and it follows that $a^x$=$b^y$,there exists $t$ € $N$ such that $a$=$t^k$,$b$=$t^q$.This lemma can be quite handy in this problem.
by aritra barua
Sat Apr 22, 2017 12:55 pm
 
Forum: Number Theory
Topic: USAJMO/USAMO 2017 P1
Replies: 3
Views: 161

Re: Beginner's Marathon

Problem $17$ : Let $\bigtriangleup$ $ABC$ be an acute triangle.$D$ is the foot of perpendicular drawn from $C$ on $AB$.Let the bisector of $\angle$ $ABC$ intersect $CD$ at $E$ and $\bigcirc$ $ADE$ at $F$.If $\angle$ $ADF$ =45˚,prove that $CF$ is tangent to $\bigcirc$ $ADE$.
by aritra barua
Wed Apr 19, 2017 6:05 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 48
Views: 1595

Re: Beginner's Marathon

The proof is trivial for some $n$ which has the form $2k$ where $k$ is an integer.Now,to say,this problem is equivalent to proving that the number of reducible fractions is even when $n$ is odd because the aforementioned sequence has precisely 'n-1' fractions(I.g:even numbered fractions).Now,we take...
by aritra barua
Tue Apr 18, 2017 9:26 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 48
Views: 1595

Re: Beginner's Marathon

Could you please assert the last line more clearly?
by aritra barua
Tue Apr 18, 2017 7:06 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 48
Views: 1595

Re: Beginner's Marathon

Problem 14:Let the incircle of triangle ABC touch BC at D. Let,DT be a diameter of the circle.If AT meets BC at M,prove that BD=CM.
by aritra barua
Tue Apr 18, 2017 6:28 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 48
Views: 1595

Re: Beginner's Marathon

Let us denote by $ABC$,the area of $\bigtriangleup ABC$. From geometric proportion,$\frac{PD}{AD} = \frac{PBC}{ABC}$ as the bases of $\bigtriangleup ABC$ and $\bigtriangleup PBC$ are the same.This case works for $\bigtriangleup PAC$ and $\bigtriangleup PAB$ as well.So,we concurr that $\frac{PD}{AS} ...
by aritra barua
Tue Apr 18, 2017 6:16 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 48
Views: 1595

Re: geomerty

<FAD itself measures 100 degree,then how can you concurr that AD=DF?I think you did not consider triangle DAF. ..According to your construction,<BCE=140 as it is supplementary to <ACB. Therefore,as CF=CD, <FCD=160 and <DFC=<DFA merely measures 10...check out your angle chasing...
by aritra barua
Sat Apr 08, 2017 12:31 pm
 
Forum: Junior Level
Topic: geomerty
Replies: 9
Views: 333
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