Popular Olympiad Stuff Questions

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*Image* Please discuss the questions Got the answer of Q3. as 837 Q4. {1,2,4} and Q6. 43 ...

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 ...

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 ...

there are 1000 men standing in a circle numbere 1 to 1000 on their back. the person with number 1 is given a sword. he kills 2 and gives sword to 3. 3 kills 4 and passes the sword to 5. the process continues so on............ ...

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 ...

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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 ...

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

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 ...

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Q Prove that the blue part is equal to the yellow part It is a square and the triangles have the intersection at the mid point of the square *Image* ...

here's a simple one prove that a number with 3^m equal digits is divisible by 3^m ...

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

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 ...

All points of the plane are colored red, blue or green. Prove that we can find two points a distance 1 apart with the same color. ...

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

At vertices of a convex polygon with 2n sides, sit hunters. while in its interior and not on any diagonal,sits a fox. The hunters shoot at the fox all simultaneously, but the fox ducks. the bullets hit the sides of the polygo ...

Q A=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+................+\frac{1}{\sqrt{10000}} Then find [A] where [] represents GINT ...

n=2p.3q how many factors of n2 are smaller than n (edit) but do not divide n? ...

Let AN be any line drawn through A in a Triangle ABC. Let BM, CN perpendiculars are drawn from B and C to AN and let D be the mid-point of BC, prove that MD = ND. ...

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} ? ...

\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)}} ...

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

find the no of positive integral solutions of \left[ \frac{x}{900}\right]=\left[ \frac{x}{901}\right] ...

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 ...

Some of the university lecturers here were interested in the question: Is it possible to divide a given square into n squares for any n≥6? (Obviously all squares neednt be the same size) ...

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

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 ...

Found somewhere.. very good problem, check how trivially you can give answer..... Is there an infinite matrix Amn such that limn→∞ amn=0 for every m and limm→∞ amn=1 for every n ? ...