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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: 84

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: 141

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: 1449

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: 1449

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: 1449

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: 1449

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: 1449

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: 311

Equation

x^2+xy+y^2=(x+y+3)^3/27,find all(x,y)
by aritra barua
Sat Mar 18, 2017 7:14 pm
 
Forum: Number Theory
Topic: Equation
Replies: 1
Views: 87

Re: IGO 2016 Elementary/4

It is just the official solution....is there any unique technique to solve that?
by aritra barua
Fri Mar 03, 2017 8:01 pm
 
Forum: Geometry
Topic: IGO 2016 Elementary/4
Replies: 8
Views: 270
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