Recently Active Olympiad Stuff Questions

Find all positive integer `n` such that the equation x3+y3+z3=nx2y2z2 has positive integer solutions. ...

In a lottery, tickets are given 9-digit nos using only the digits 1,2,3.They are also coloured red, blue or green in such a way that 2 tickets whose numbers differ in all the 9 places get different colours! Suppose the ticket ...

I look at askiitians.com only when I am VERY VERY BORED. But one post took me quite by surprise, when the student asked, if f:N→N is a strictly increasing function satisfying f(f(n)) = 3n for all n, find f(11). ...

Prove that for any n, \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...... + \frac{1}{n^2}<1 Note:This is not a very strict inequality since the exact value of this limit as n goes to infinity is 0.645 ...

'z' is a complex number satisfying za+zb+zc+1=0. Given a,b,c are distinct integers..... Proe |z|=1. From Shastra ...

Sagnik just posted this one on my chatbox... Prove that (a,b, c +ve) a2+1/b+c + b2+1/a+c + c2+1/a+b ≥3 RMO: 2006 Hint: Nesbitts? ...

Show that there exist infinitely many positive integers A such that (1) 2A is a perfect square ; (2) 3A is a perfect cube ; (3) 5A is a perfect fifth power. ...

This turned up in 1 of the RMO's of WB region, I found it nice while solving, so flicked it here - Prove the sum 1/1001 + 1/1002 +.... 1/3001 lies between 1 and 4/3 . ...

Hello! The NSEA will be held soon and I wanted some sample questions. Please help. ...

If a+b+c = 2, prove that 9abc+8 ≥ 8(ab+bc+ca) ...

An ex-goiitian and present iitian is an orgainizer for an online math competition named Shaastra. The first question is a nice inequality: If x,y,z \in \mathbb{R^+} are such that x+y+z = 9(xyz)^{\frac{2}{3}} prove that \frac{ ...

Is it possible to construct a continuous and differentiable curve in the cartesian plane such that both the co- ordinates can never be simultaneously rational ? take 0 < x,y <1 ...

prove for all integral n, 2n >= 1 + n.2(n-1)/2 ...

Schauspiel Solutions is a technological company which provides solutions to almost any technical problem. Considering the uniqueness of the name, and being headed by a very brilliant CEO, the company gathers a huge deal of pu ...

This will be torn apart, still it should help us recall some useful facts: Prove that: \sum_{cyc} \frac{(a+b)(a+c)}{(a-b)(a-c)} = 1 ...

Figure out integral solutions of x3-y2=2 ...

1) Find the minimum of (a+b)4+(b+c)4+(c+a)4- 4/7 (a4+b4+c4) a,b,c are reals. 2) Solve the Diophantine equation for integers x3+2y3+4z3-6xyz=1 Beleive me, number 2 is really hard... ...

a,b,c are positive reals such that abc=1 prove *Image* ...

Use the Pigeon-Hole Principle to derive Euler's totient function....[62] ...

If m is a prime number,and a,b ,two numbers less than m prove that am-2+am-3b+am-4b2+...bm-2 is a multiple of m ...

If φ(N) is the number of integers which are less than N and prime to it and if x is prime to N then show that xφ(N)-1≡0.mod(N) ...

Show that ax+a and ax-a are always even whatever a and x may be I did it by taking four cases for a and x [EE,EO,OE,OO] where E and O denote even and odd respectively Is any shorter method available ? ...

If p is prime ,and x is prime to p,show that xpr-pr-1-1 is divisible by pr ...

If p and q are any two positive integers ,show that (pq)! is divisible by (p!)q.q! and by (q!)p.p! ...

Find general solution of congurence 98x-1≡0(mod 139) ...

Show that a12-b12 is divisible by 91 if a and b are both prime to 91 ...

If p is a prime number and a is prime to p,and if a square number c2 can be found such that c2-a is divisible by p, then show that a(p-1)/2-1 is divisible by p ...

find the largest integer n such that n+5 divides n3+25 ...

Determine whether there exist rationals (x,y) such that x2+y2=3 ...

show that n! < {(n+1)/2}^2 for n belonging to N ...