Recently Active Olympiad Stuff Questions

i am from 10 std. i got 35 marks in rmo and was selected for inmo on 6th feb 2011.how should i prepare?please help ...

Q:=> \textup{Prove that amongst 10 successive natural nos. there are always atleast} \textup{one and atmost four numbers that are not divisible by any of the numbers 2,3,5,7} ...

1)prove that (2+√5)1/3 + (2-√5)1/3 is rational. 2)Let a,b,c be distinct real numbers.prove that (a-b)1/3+ (b-c)1/3+ (c-a)1/3≠0 ...

1) Prove that nn> 1.3.5.....(2n-3)(2n-1) 2) Let a, b, c be the sides of a triangle with perimeter 2. Prove that a2+b2+c2+2abc<2 3) -5<x<11. What is the greatest value of (11-x)3(x+5)5 4) x,y,z are positive real nu ...

] On an in finite chessboard (whose squares are labeled by (x; y), where x and y range over all integers), a king is placed at (0; 0). On each turn, it has probability of 0.1 of moving to each of the four edge- neighboring sq ...

How many functions f from {−1005, . . . , 1005} to {−2010, . . . , 2010} are there such that the following two conditions are satisfied? • If a < b then f(a) < f(b). • There is no n in {−1005, . . . , 1005} su ...

Let p(n) denote the number of ways in which n can be written as a sum of positive integers. (order of the summands being irrelevant). prove that p(5k+4)≡0 (mod 5) where k is any positive integer. ...

In this page: http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem In the Second method proof that is given there: Can someone visit the page and tell me why should it be so that when a,2a,3a,.....(p-1)a when divi ...

find the coeffcient of (x^a)(y^b) in the expansion (x+y+xy+2)^(2008), where both a and b are even nos? ...

Find all prime numbers that satisfy simultaneously p+1=2x2 p2+1=2y2 I solved it by the most elementary method can u all do the same even to prophet Sir(its not a challenge but the simplest solution actually doesnt require num ...

hey friends.... can u plz post some qstns asked in previous years GMO.... i cud nt get frm any site..... plzzzzzzzzzzz..... ...

Let S denote the set of all triples (i, j, k) of positive integers where i + j + k = 17. Compute : SUMMATION (x*y*z) , for all (i,j,k) belonging to set S ...

How many functions f : {1, 2, 3, 4, 5} to {1, 2, 3, 4, 5} satisfy f(f(x)) = f(x) for all x= {1, 2, 3, 4, 5} ? ...

*Image* ABCD is a prallelogram. Vertex D meets the side BC when produced to E such that LDEC = 90° and LDCB = 150°. A person starts from E and reaches C with some speed in time t seconds and then from C to B and again from ...

1)in an equilateral triangle of side a, the distance of a point P from its vertices is given 3.4 and 5 respectively.FIND THE VALUE OF THE SIDE a. 2)On similar ground try to to find the side of a square ABCD such that the dist ...

What is the remainder when 21991 is divided by 1991 ...

Let a be the smallest +ve integer with the property as follows: there are distinct positive integers a1,a2,a3,a4,a5 such that g(x)=(x-a1)(x-a2)(x-a3)(x-a4)(x-a5) has at least `a` non zero coefficients find set of (a1,a2,a3,a4 ...

An equilateral triangle is divided into 36 smaller equilateral triangles by dividing each side into 6 parts and joining the corresponding points on adjacent sides of the triangle. *Image* There is a drunk ant at each of the 2 ...

Show that the number 510-310 is divisible by 11 ...

Well, Actually I don't know anything on how to solve such type of problems. Can anyone suggest any article or make a post for better understanding on how to approach such problems. Thanks! ...

a3+b3+c3+3abc≥ab(a+b)+bc(b+c)+ca(a+c).....a,b.c +ve reals ...

1 > Solve the following equation for " x " - 16 x 6 - 24 x 4 + 10 x 2 - 1 = 1/8 x 2 > The positive reals " x " , " y " , " z " satisfy the following equation - x 2 + x y + y 2/3 = 25 y 2/3 + z 2 = 9 z 2 + z x + x 2 = 16 ...

Let a1=0 and a2=1. an= n/2 an-1+ n/2 (n-1)an-2+(-1)n(1- n/2 ). Find f=an+2 nC2an-1 +3 nC3an-2+...+na1. ...

Prove that for all real a,b and c satisfying: a^3+b^3=c^3 the following inequality holds: a^2+b^2-c^2> 6(c-a)(c-b) ...

Prove that 1/1001 + 1/1002 + 1/1003 +...................+ 1/3001 lies b/w 1 & 4/3 ...

Find ax5 + by5 where a,b,x,y are real numbers satisfying ax+by = 3 ax2+by2 = 7 ax3+by3 = 16 ax4+by4 = 42 ...

Find all the pairs of positive numbers such that the last digit of their sum is 3, their diference is a primer number and their product is a perfect square. (a previous olympiad question,interesting one) ...

Q 1 A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in length. What is the maximum area he can enclose by connecting the pieces? Assume the land is flat. ans= 2 6 yards ...

*Image* *Image* EDITED - It indeed is " Sophomore's Dream " , as Sir has mentioned below . I actually got the result in a different way than what is done in many sites , so that ' s why I posted the quesion . ...

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