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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: 67

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: 66
Views: 2178

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: 66
Views: 2178

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: 66
Views: 2178

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: 8
Views: 204

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: 350

Re: MPMS Problem Solving Marathon

$Problem 4$: If $7^m$+$11^n$ is a perfect square,then $m$ must be even and $n$ must be odd. (By using mod $3$ and mod $4$);let $m$=$2q$ and $n$=$2k$+$1$ and $49^q$+$121^k.11$=$a^2$ $\Rightarrow$ ($x$+$7^q$)($x$-$7^q$)=$121^k$.$11$.Now make a total of $2$ cases along with $6$ subcases each.$1$ case w...
by aritra barua
Mon Jun 05, 2017 1:47 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 350

Re: MPMS Problem Solving Marathon

$Alternative Solution to 3$:It is obvious that if $4n^2$-$6n$+$45$ is a perfect square,then $4n^2$-$6n$+$45$ $\equiv$ $1$ (mod $2$) $\Rightarrow$ $4n^2$-$6n$+$45$ $\equiv$ $1$ (mod $4$) $\Rightarrow$ $1$-$6n$ $\equiv$ $1$ (mod $4$) $\Rightarrow$ $n$ $\equiv$ $0$ (mod $2$);so $n$ cannot be odd.
by aritra barua
Mon Jun 05, 2017 1:17 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 350

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: 8
Views: 350

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: 99
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