Recently Active Olympiad Stuff Questions

For any natural number n≥2, find out which of the real positive numbers a and b is greater, knowing that an=a+1 and b2n=b+3a. ...

This time's Pan-African Olympiad had a problem that goes: If, 1 \le i \le 2000, \ x_i \in \{-1,1\} is it possible to have x_1 x_2 + x_2x_3+x_3x_4+....+x_{2000} x_1 = 999 ...

When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one) to the entire complex plane (sans simple pole at s = 1). Riemann noted that his zeta ...

This is a little bit easier than the previous Induction qn: The problem may be familiar to some - it is from an old Hungarian Olympiad (a1,a2,a3,...,an) is a permutation of (1,2,3,...,n). n is an odd integer Prove that (a1-1) ...

As the heading says, I am looking for an MI proof only for this: (its relatively easy otherwise) p≥3 is an integer and a and b are roots of x2-(p+1)x+1. Prove that for all natural numbers n, an+bn is not divisible by p. ...

Find out the remainder of 19^{92}/92 ...

Let s1, s2, s3 be 3 tangent spheres each with each. These spheres are placed on a table, and their radii are 1,2,3 respectively. A plane tangent to all these spheres is considered. What is the angle made by this plane with th ...

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colore ...

Is there a sequence of positive integers in which every positive integer occurs exactly once and for every k = 1, 2, 3, . . . the sum of the first k terms is divisible by k? ...

In the plane there are n points such that of any 3 points always two points can be chosen such that their distance is less than 1. Prove that within these n points there are [n+1/2] which can be covered by a circular area wit ...

Camp for Freshmen... N boys studying physics and N girls studying liberal arts are sitting by the flickering campfire. Just by them are N, strictly coed tents (for two people each). Egon Quark has to arrange the sleeping orde ...

If a and b are positive integers such that for all positive integers k, ak| bk+1, then prove that a|b ...

If a,b,c are sides of a triangle, prove that \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} ...

Let P(x) be a nonzero polynomial such that, for all real numbers x, P(x^2 - 1) = P(x)P(-x). Determine the maximum possible number of real roots of P(x). ...

A conference has 47 people attending. One woman knows 16 of the men who are attending, another knows 17, and so on up to the m-th woman who knows all n of the men who are attending. Find m and n. ...

Given a regular 2007-gon. Find the minimal number k such that: Among every k vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the ...

Q. 1. Find all polynomials p(x) such that x p(x-1) = (x-15) p(x) . Q. 2. Determine all functions f : R - {0,1} --> R such that *Image* Thank You ! [111] ...

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: (i) The first fly is caught after a resting period of 1 minute. (ii) The resting period before the 2mth fly ...

Here are the Questions i could remember ..... Q.1. What should be the values of x and y such that x2 + y2 is minimum and (x + 5)2 + (y – 12)2 = 142 ?? I got the answer as (x, y) = ( 5/13, -12/13 ) and min value as 1 Q.2. so ...

Prove that 2mn> mn as well as nm Find the shortest proof. HINT : this is in syllabus oops sry, condition: n,m>0 and ε N ...

let 'x' be positive non-square integers such that x1=2, x2=3, x3 = 5, x4=6 .. and so on... and we define <m> such that if the decimal part is <=0.5 we have <m> =m and if decimal part >0.5 , then <m>=m+ ...

Could anyone please post a copy of all B.Stat Subjective Questions? ...

Q1. Let ABC be a triangle and let P be an interior point such that <BPC=90°, <BAP=<BCP. Let M,N be the midpoints of AC,BC respectively. Suppose BP=2PM. Prove that A,P,N are collinear ...

What is the largest positive integer n so that n! can be expressed as the product of n - 3 consecutive positive integers? ...

8^{\frac{1}{n}} + 27^{\frac{1}{n}} + 64^{\frac{1}{n}} > 12^{\frac{1}{n}} + 32^{\frac{1}{n}} + 36^{\frac{1}{n}} for n>1 ...

prove that there are no positive integers m and n such that m(m+1)(m+2)(m+3)=n(n+1)^{2}(n+2)^{2}(n+3)^{2} ...

Prove that, for any integers a, b, c, there exists a positive integer n such that \sqrt{n^{3}+an^{2}+bn+c} is not an integer. ...

*Image* given AB=L1 AO=L2 E is mid point of arc BAEC EO perpendicular to AC find OC .. a starter .. ...

Show that the interval [0, 1] cannot be partitioned into two disjoint sets A and B such that B = A + a for some real number a. ...

Find the least positive integer n such that any set of n pairwise relatively prime integers greater than 1 and less than 2005 contains at least one prime number. ...