Recently Active Olympiad Stuff Questions

Evaluate: \int_{0}^{\infty}{e^{-\alpha^2\left(x^2+\frac{\beta^2}{x^2} \right)}}dx You may use: \int_{0}^{\infty}{e^{-x^2}dx}=\frac{\sqrt{\pi}}{2} ...

Mr and Mrs. Bajaj invite three other couples to a party and as usual some people shake hands. Spouses do not shake hands with each other. At the end of introductions, Mr. Bajaj asks all others including Mrs. Bajaj how many ti ...

Show that in any set of eleven integers there are six whose sum is divisible by 6 . ...

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others. ...

1 problem from FMO. Given that \sqrt{n}+\sqrt{n+2005} is an integer. Find all integers n for which it is true. ...

Given 2 positive integers a,b satisfying a< b . Prove that among every b consecutive positive integers there are two numbers whose product is divisible by ab . ...

In an international meeting of n≥3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken more that by the half of the participants. What is the least valu ...

a man is left in a field where a point O is placed.there r two flags named A,B .he is told that if he moves from O to A and the same distance again at an angle 90 from A he reaches A'. and similarly if moves from O to B and t ...

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although m not the right person to give problem for practice but still m posting some ques for practice.....i jus started this cozz i thinkj the threads r still in the joy nd shock of jee results:P:D.. 1)In a book with page n ...

a≥b>0=> a+a/b(a-b)≥4 hey its a/b.1/(a-b) ...

x+y=2 =>x3 y3 (x3+y3)≤2 x,y +ve reals ...

Prove that there exists an irrational r for every natural number k(>=2) such that [rm]≡-1mod(k) for every natural number m. Here [x] is the greatest integer less than or equal to x. ...

\sum_{n= 1}^{n=\infty }{}\frac{1}{\left\{(2n- 1 )^2- ( 2m ) ^2 \right\}^2} 2. \sum_{n=1}^{n= \infty}{\frac{1}{n( 36n^2-1)}} 3. \sum_{n=1}^{n= \infty}{\frac{1}{n( 9n^2-1)}} ...

Suppose positive integers m, n, K satisfy mn = K2 + K + 3. Prove that at least one of the following Diophantine equations x2+11y2=4m and x2+11y2=4n has a solution (x, y) with x, y being odd numbers EDITED sorry for the incorr ...

Find all natural numbers n > 1 such that n2 does not divide (n − 2)!. ...

Find all positive solutions to the following Diophantine equation ------ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 + h 4 = 26959 . ...

suppose a and b are real numbers such that the roots of the cubic equation ax^3-x^2+bx-1 =0 are all positive real numbers. Prove that: (i) 0<3ab\le1 and (ii) b\ge \sqrt{3} ...

Let a,b,c,x,y,z be reals and let A=ax+by+cz . Similarly B=ay+bz+cx & C=az+bx+cy . Given |A-B|\ge 1 , |B-C|\ge 1 & |C-A|\ge 1 . Find the minimum of (a^2+b^2+c^2)(x^2+y^2+z^2) . [54] ...

I remember Nishant Bhaiya's face when he was discussing this problem....well he hates this type of abstract problems but i love this one.... find the value of: \sum_{1}^{\propto }{(1/i)} well i warn sum of H.P. has no formula ...

Find all pairs (x, y) of integers such that x^3 - y^3 = 2xy + 8 ...

prophet sir .. have a try at this one :) Each of the boys A and B tells the teacher a positive integer but neither of them knows the other`s number.The teacher writes 2 distince positive integers on the blackboard and announc ...

Here are two good questions from Mathlinks , maybe you should give it a try . 1 > Let the positive integers a 1 , a 2 , a 3 , a 4 .... satisfy this eqn . , { 1 / a 1 } + { 1 / a 2 } + .... { 1 / a 100 } = 20 . Prove that t ...

prove that bc(b+c)+ca(c+a)ab(a+b)<=2(a3+b3+c3) there r many sol to this sum but 1 ol just takes 2 steps lets see who get it ...

This is a brilliant brilliant proof i read today... *Image* This is from a book called problem solving strategies... (it is a brilliant read!) ...

8 pieces are placed on a chessboard such that no two pieces lie on the same row or column. Prove that an even number of pieces lie on black squares ...

There's a convex quadrilateral....Show that if we can inscribe 2 squares in the quad, then we can inscribe infinite numbers of squares - all of different lengths. A square is said to be inscribed in a quad, if the vertices of ...

If the hr and minute hands of a clock are interchanged - in how many cases shall the new setup show some time? ...

if p is a prime no satisfying n<p<2n show that 2n C is divsible be p............... n ...

There is a cone with height h & radius r. An insect at the bottom edge of the con needs to complete the journey around the axis of the cone. Find the shortest distance the insect has to cover. *Image* this is the simplest pro ...