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Re: Beginner's Marathon

Problem $26$
Given an acute angled triangle $\bigtriangleup ABC$.Inscribe in it a triangle $UVW$ of least possible perimeter.
by aritra barua
Fri Jul 07, 2017 1:08 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3616

Re: More primes and more primes!

Apparently,this is a TST problem from Hong Kong.We rewrite our equation:$2^m*p^2$=$y^7-1$=$(y-1)(y^6+y^5....+y+1)$.Clearly $y$ being odd implies $y^6+y^5...+y+1$ is odd.Hence $2^m$ must divide $y-1$ since $p$ is a positive integer.Now the equation $\frac{y-1}{2^m}$*$(y^6+y^5...+y+1)$=$p^2$ leads us ...
by aritra barua
Fri Jul 07, 2017 10:45 am
 
Forum: Number Theory
Topic: More primes and more primes!
Replies: 1
Views: 234

More primes and more primes!

Find every triplet of positive integers ($m,p,y$) such that the equation:$2^m*p^2+1$=$y^7$ is satisfied when $p$ and $y$ are primes.
by aritra barua
Wed Jul 05, 2017 7:55 pm
 
Forum: Number Theory
Topic: More primes and more primes!
Replies: 1
Views: 234

Infinite solutions

Find with proof every pair of ($a$,$b$) in general form such that $a+b^2$|$a^3+b^3$.
by aritra barua
Tue Jul 04, 2017 7:46 pm
 
Forum: Number Theory
Topic: Infinite solutions
Replies: 1
Views: 209

IMO 2009 SL(G-1)

Let $\bigtriangleup$ $ABC$ be an isosceles triangle with $AB$=$AC$.The bisectors of $\angle BAC$ and $\angle ABC$ intersect $BC$ and $AC$ at $D$ and $E$ respectively.Suppose $\angle BEK$ to be $45$ degree,where $K$ is the incenter of $\bigtriangleup$ $ACD$.Determine all possible values of $\angle BA...
by aritra barua
Mon Jun 19, 2017 3:02 pm
 
Forum: Geometry
Topic: IMO 2009 SL(G-1)
Replies: 1
Views: 216

Re: Beginner's Marathon

What has been meant by 'after 2 hours 30 min'?
by aritra barua
Thu Jun 15, 2017 2:44 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3616

Re: Beginner's Marathon

An easy problem:$\text{Problem 20}$
The fractions $\frac{7x+1}{2}$, $\frac{7x+2}{3}$,......., $\frac{7x+2016}{2017}$ are irreducible.Find all possible values of $x$ such that $x$ is less than or equal to $300$.This Problem was recommended by Zawad bhai in a mock test at CMC.
by aritra barua
Fri Jun 09, 2017 4:27 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3616

Re: Beginner's Marathon

$\text{Problem 19}$ Let,the number of games played by a person $i$ be $g_i$.Therefore,the sum $g_1$+$g_2$+......+$g_{127}$ is even as each game is counted twice.We,assume that the number of people who played an odd numbered games is odd.In that case,if we eliminate their played games from the sum,we...
by aritra barua
Thu Jun 08, 2017 11:42 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3616

Re: 2017 Regional no.9 Dhaka

Check the parity of the equation and the rest should be clear.
by aritra barua
Mon Jun 05, 2017 2:31 pm
 
Forum: Divisional Math Olympiad
Topic: 2017 Regional no.9 Dhaka
Replies: 9
Views: 480

Re: MPMS Problem Solving Marathon

Simplier solution to problem $1$,I used this after getting to know what Wilson's Theorem is from dshasan's post.Let, $a$=$1$ . $2$...... $2k$.Telescope $a^2$ as $1$ . $2$.... $2k$.($-2k$)....($-2$)($-1$).From the telescoping,it follows $a^2$ $\equiv$ $1$.$2$....$2k$($p-2k$)....($p-2$)($p-1$) (mod $p...
by aritra barua
Mon Jun 05, 2017 2:09 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 594
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