Search found 186 matches

by photon
Sat Feb 22, 2014 10:12 pm
Forum: Number Theory
Topic: infinite primes
Replies: 4
Views: 1452

Re: infinite primes

What's wrong ? :| :?: By Fermat's little theorem and as given , $pq|2^{p-1}-1$ and $pq|2^{q-1}-1$ . Let $2^{p-1}=pqc+1$ , $2^{p-1}=pqd+1$ . ( $c,d$ are 2 positive odd integers .) WLOG , $p<q$ . $2^{p-1}+2^{q-1}=2^{p-1}(2^{q-p}+1)$ $\Rightarrow pq(c+d)+2=2^{p-1}(2^{q-p}+1)$ , $\Rightarrow pqm+1=2^{p-...
by photon
Wed Feb 19, 2014 1:17 pm
Forum: Algebra
Topic: x,y,z>1
Replies: 2
Views: 1045

x,y,z>1

Prove that if $x,y,z$ $>$ $1$ , and $\displaystyle \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$ , then
$\displaystyle \sqrt{x+y+z}\geq\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$

Source - IMO longlist (1992)
by photon
Wed Feb 12, 2014 9:19 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

Re: warm-up problems for national BdMO'14

Problem 12. Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that ...
by photon
Wed Feb 12, 2014 8:08 pm
Forum: Geometry
Topic: Touching Circumcircles around Incentre [Self-Made]
Replies: 4
Views: 1334

Re: Touching Circumcircles around Incentre [Self-Made]

$\angle MKB=\angle KBC= \frac{1}{2}\angle B=\angle MBK $ , so $MB=MK$ . As in $\Delta AKB$ , $MA=MK=MB$ , $M$ is the center of the circumcircle of $\Delta AKB$ . M is the midpoint of AB , so it is a right-angle triangle - $\angle AKB=90^o$ . Similarly , $\angle ALC=90^o$ . $\angle AKI+\angle ALI=180...
by photon
Wed Feb 12, 2014 5:17 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

Re: warm-up problems for national BdMO'14

Solution of 4 : circumcircle of $\Delta ADB$ cuts $AC$ at $Q$ . $\angle QBD = \angle QAD = 36^o$ , $\angle QBC + \angle CBD = 36^o \Rightarrow \angle QBC = 18^o $ , similarly $\angle QDC = 36^o$ . As $DC $ and $BC $ are angle bisector of $\angle QDP$ and $\angle QBP$ respectively ; $\displaystyle \f...
by photon
Tue Feb 11, 2014 10:29 am
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

Re: warm-up problems for national BdMO'14

problem 10 was stated wrong , now it is edited .
by photon
Mon Feb 10, 2014 9:18 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

Re: warm-up problems for national BdMO'14

8. $$\binom{n}{n-1}n=3n$$ So, $$n=3$$ This could be written when n is variable , not necessarily true for constant $n$ . :? (Though answer is correct) My solution to problem 8 $2n^k+3n=(n+1)^n-1=n [ (n+1)^{n-1}+(n+1)^{n-2}+..............(n+1)^1+1 ]$ $2n^k+3n \equiv n[(n+1)^{n-1}+(n+1)^{n-2}+..........
by photon
Mon Feb 10, 2014 9:17 pm
Forum: Algebra
Topic: simple equation
Replies: 3
Views: 2886

Re: simple equation

This problem asks any value of $x$ . No integer solution is possible , but fraction , irrational solution can be possible .
by photon
Sun Feb 09, 2014 7:10 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

Re: warm-up problems for national BdMO'14

@Sowmitra , I had solved this using totient theorem ,but $7^4$ made it clean . Though it was $7^{1997}$ , doesn't matter ... @Fatin , it is not $LHS\not=RHS$ , it is that considering $\lambda_n$ as integer the equation got wrong , so contradiction for integer :) . $\lambda_n \notin \mathbb{N}$ [$\la...
by photon
Sun Feb 09, 2014 1:12 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

warm-up problems for national BdMO'14

This thread is for some practice as National Math Olympiad is knocking at door . Others may get benefitted , share and learn things together from here . One thing should be clear that this is not just about to get prize- you try , solve , learn and improve your skill and most important is having fun...