Recently Active Olympiad Stuff Questions

For any integer k>1, prove that there is exactly one power of 2 having exactly k digits with leading digit 1 (when written in decimal system). ...

find out a four digit number M such that N=4xM and N has the following properties : i) it is also a four digit no. ii) the digits of N are in reverse order of those in M. ...

Here is the collection of all problems... http://pertselv.tripod.com/RusMath.html Enjoy.......[6] ...

In a \Delta ABC , let x=\tan\left(\frac{B-C}{2}\right)\tan\left(\frac{A}{2}\right) y =\tan\left(\frac{C-A}{2}\right)\tan\left(\frac{B}{2}\right) z =\tan\left(\frac{A-B}{2}\right)\tan\left(\frac{C}{2}\right) Prove that x+y+z+x ...

Solve for natural 'k', x2+y2+z2=kxyz, x,y,z are naturals. ...

prove that if 2n - 1 is prime then n is also prime ...

It is not of this year... ABc is a triangle with AB=AC...D is the mid-point of BC, P is any other point on AD. PE is perpendicular to AC. \frac{AP}{PD}=\frac{BP}{PE}=\lambda \\ \ \frac{BD}{AD}=m\\ \ z=m^2(1+\lambda)\\ \text { ...

My friend gave me this question.. For n\ge2 Prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer. ...

A row contains 1000 integers The second row is formed by writing under each integer, the number of times it occurs in the first row The third row is now constructed by writing under each number in the 2nd row, the number of t ...

For how many positive integers n is (1999+ 1/2)n + (2000+ 1/2 )n an integer? ...

Prove that for all n\ge1 , 1+5^n+5^{2n}+5^{3n}+5^{4n} is composite ...

{\color{red} 17^{14}>31^{11}-\mathit{Prove...}} ...

Two positive integers are chosen. The sum is revealed to logician A, and the sum of squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation b ...

x^3+y^4=7 Prove that the eqn has no integer solns. ...

India's rank :- 28! Cheers! Our mates have done quite a gr8 job! Detailed results avilable at :- http://www.imo-official.org/year_info.aspx?year=2009 ...

find all ordered triples (x,y,z)of real numbers which satisfy the following system of equations: xy = z -x - y xz = y - x - z yz = x - y - z ...

Prove that x9999+x8888+x7777+.....+x1111+1 is divisible by x+1 ...

If Sn is the sum Sn = 1 + 1/2 + 1/3 + ... 1/n ( n >2 ) , prove that *Image* ...

Prove sin10 is irrational. ...

Let A be the sum of the digits of the number 4444^{4444} , and B the sum of the digits of A. Compute the sum of the digits of B. ...

Its nice to see so much of action in this section in particular, and in the forums in general, but I have a suggestion about contributing problems. I can sum it up as: Be Genuine My feeling is that we can all truly enjoy and ...

Prove that there are infinitely many primes congruent to 9 mod 10. ...

In spite of the spate of unanswered questions, I would like to add one more question to the forum: Find all n such that 2n divides 3n+1 (Hope this question too does not end up as a statistic) edited ...

If 2n+1 and 3n+1 are both squares then prove that (n is a natural number): 1. 5n+3 is not a prime 2. n is divisible by 40 ...

The first day of problem-cracking at the 50th International Math Olympiad will be underway in a few hours from now. It is being held at Bremen, Germany (I had stayed there for three months for my IIT project work :D). You can ...

A positive integer n is a good number if it can be written as the sum of two positive integers a and b, i.e. n=a+b, such that n|ab. It is a very good number if a,b are distinct. Find all numbers n such that n is good but it's ...

Determine all pairs of positive integers (a,b) such that all roots of x^3+8x^2+2^ax+2^b = 0 are real and at least one root is an integer. ...

This appeared in one of the RMOs: Solve for real x and y: 5x\left(1+\dfrac{1}{x^2+y^2}\right)=12 5y\left(1-\dfrac{1}{x^2+y^2}\right)=4 ...

Find the real roots of the equation: x^3+2ax+\dfrac{1}{16}=-a+\sqrt{a^2+x-\dfrac{1}{16}} where 0<a<\dfrac{1}{4} Edit: The power of x is 2 and not 3 (thanx gordo for pointing it out). And I placed it mistakenly in this s ...

let d(n) denote no. of divisors of n,then prove that d(n)<2 n ...