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NMTC 2012 ( CLASS 9th and 10th) Question paper


The AMTI is a pioneer organisation in promoting and conducting Maths Talent Tests in India. Last year (43rd TC Data) (in the 43rd National level tests) 54058 students from 332 institutions spread all over India, participated at the screening level; 10% of them insitutionwise were selected for the final test. For the benefit of final level contestants and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students.

DOWNLOAD THE 2012 PAPER :
NMTC Question Paper

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CGMO – 2012 ( China Girls Math Olympiad 2012 ) Problem 7


Let  \{a_n\} be a sequence of nondecreasing positive integers such that  \frac{r}{a_{r}}= k+1 for some positive integers k and r. Prove that there exists a positive integer s such that  \frac{s}{a_s} = k

Infinite number of primes :)


In today’s post , I will present a very well known proof. What makes this proof especially appealing is that it is not too complex. Even so, it is very powerful. This theorem was first presented in Euclid’s Elements (Book IX, Proposition 20).

Theorem: There are an infinite number of primes.

Proof:

(1) Assume that there is only a finite number of primes.

(2) Then, there exists a prime pn that is the largest prime.

(3) Let p1, p2, …, pn be the list of all primes that exist.

(4) Let x = p1*p2*…*pn + 1.

(5) By the fundamental theorem of arithmetic (see Theorem 3, here), we know there is at least one prime that divides x. Let us call this prime p*.

(6) But none of the primes p1 … pn divide x since x ≡ 1 (mod pi) for any of the primes p1 … pn

(7) Therefore, we have a contradiction. We have a prime p* that is not in the complete list of primes.

(8) So, we reject our assumption in step#1 and conclude that there are an infinite number of primes that exist.

QED

Largest prime number ever know !


The largest prime number till found in

243112609  –  1   ( mersenne primes )
This number was discovered  by G1 year 2008 . This number has total of 12978189 digits .   Landon Curt Noll computed the decimal value of this prime with calc and the English name with number.

Here is the top 10 mersenne primes known

rank prime digits who when reference
1 243112609-1 12978189 G10 2008 Mersenne 47??
2 242643801-1 12837064 G12 2009 Mersenne 46??
3 237156667-1 11185272 G11 2008 Mersenne 45?
4 232582657-1 9808358 G9 2006 Mersenne 44?
5 230402457-1 9152052 G9 2005 Mersenne 43?
6 225964951-1 7816230 G8 2005 Mersenne 42?
7 224036583-1 7235733 G7 2004 Mersenne 41
8 220996011-1 6320430 G6 2003 Mersenne 40
9 213466917-1 4053946 G5 2001 Mersenne 39
10 26972593-1 2098960 G4 1999 Mersenne 38

THANK YOU !