## Search found 186 matches

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

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 , $p<q$ . $2^{p-1}+2^{q-1}=2^{p-1}(2^{q-p}+1)$ $\Rightarrow pq(c+d)+2=2^{p-1}(2^{q-p}+1)$ , $\Rightarrow pqm+1=2^{p-... Wed Feb 19, 2014 1:17 pm Forum: Algebra Topic: x,y,z>1 Replies: 2 Views: 1045 ### x,y,z>1 Prove that if$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}.$Source - IMO longlist (1992) Wed Feb 12, 2014 9:19 pm Forum: National Math Olympiad (BdMO) Topic: warm-up problems for national BdMO'14 Replies: 25 Views: 5655 ### Re: warm-up problems for national BdMO'14 Problem 12. Let$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$. Show that ... Wed Feb 12, 2014 8:08 pm Forum: Geometry Topic: Touching Circumcircles around Incentre [Self-Made] Replies: 4 Views: 1334 ### Re: Touching Circumcircles around Incentre [Self-Made]$\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...
Wed Feb 12, 2014 5:17 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 5655

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... 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 . 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}+..........
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 .
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...
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...