Recently Active Olympiad Stuff Questions

let a 602 digit number 111100111100111100.........................11110013 find the remainder when it is divided by 13 ...

Maybe this is pretty easy -- but I liked it a lot -- so here goes -- Find the remainder when 1003 x 1009 x 1017 is divided by 119. ...

If 4x-y is divisible by 3, prove that 4x2+7xy-2y2 is divisible by 9. ...

This is fun to work out. Let \tau(m) represent the number of divisors of the natural number m. Then prove that \tau(1)+\tau(1)+ ...+\tau(n) = \left[\frac{n}{1} \right] + \left[\frac{n}{2} \right]+...+\left[\frac{n}{n} \right] ...

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

Could anyone post the questions please? I am told it was held today. ...

find the least positive integer m such that 22000 divides 2003m-1 ...

If xy+yz+zx = 3, prove that \sqrt{1+x^4} + \sqrt {1+y^4} + \sqrt {1+z^4} \ge 3 \sqrt 2 ...

Solve for positive reals (x,y,z) z^2+2xyz=1 3x^2y^2+3y^2x=1+x^3y^4 z+zy^4+4y^3=4y+6y^2z It's been a while no -one has really given any elaborate soln for this..... ...

do there exist 1 000 000 positive integers such that the sum of any collection of these integers is never a perfect square. ...

*Image* Suppose ABC is an equilateral triangle BD/BC = 1/3 CE/CA = 1/3 AF/AB = 1/3 find the area of the shaded triangle divided by the area of triangle ABC. ...

It is not at all hard , ( not only from the view point of sirs ) , so try this ---- 1 > Let a , b , c, d , e …… be the positive divisors of “ n “ except n and 1 . Prove that --------- ( 1 / a ) + ( 1 / b ) + ( 1 / ...

If x and y are integers such that (x+2y)2 + (x+4y) = 710 The value of x is (A) 13 (B) 15 (C) 18 (D) 26 ...

(1) Let ABC be a triangle with AB =3, BC =4 and CA =5. A Line L,which is perpendicular to AC,Intersects AC in Q and AB in P.Suppose there is a Circle inside the Quadrilateral PBCQ touching all its four sides (i.e, PBCQ has an ...

This one appeared in one of the Russian Olympiads. Find the final five digits of the number N = 999...9 that contains 1001 nines positioned as above. ...

find all odd prime numbers p which divide 1p-1+2p-1+3p-1+.....................2004p-1 ...

This one was once asked (2yrs ago) by one of my friends.... Prove that among 39 sequential natural numbers there always is a number with the sum of its digits divisible by 11..... ...

These are few sums on probability from Chinese Olympiad.. 1. If a stick is broken in two at random,what is the average length of the smaller piece ?? 2. A railroad numbers its locomotives in order, 1,2,....N. One day you see ...

If x=\left(16^{3}+17^{3}+18^{3}+19^{3} \right) , then x divided by 70 leaves a remainder of what? ...

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) ...

i was unable to solve this one but wait till you see the solution given in the book!! if a,b,c,d are four non-negative reals and a+b+c+d=1, show that ab+bc+cd \le \frac{1}{4} ...

1.Determine whether or not there exists a positive integer n such that n is divisible by exactly 2000 different prime nos,and 2^n+1 is divisible by n. ...

The figure here shows a 3 x 3 grid. As you can see each cell can have maximum four walls. What is maximum number of walls that a N x N grid can have? This 3 x 3 grid has 24 walls. *Image* ...

\left(a+b+c \right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)=16 , a,b,c are strictly positive reals......Maximise and minimise \frac{a}{b}+\frac{b}{c}+\frac{c}{a} ... ...

\int_{-1}^{0}{\int_{-\sqrt{1-x^{2}}}^{0}{\frac{2}{1+\sqrt{x^{2}+y^{2}}}}}dydx ...

Prove that: \begin{vmatrix}\frac{1}{p+1}&\frac{1}{p+2}&\ldots &\frac{1}{p+n}\\ \frac{1}{p+2}&\frac{1}{p+3}&\ldots &\frac{1}{p+n+1}\\ \vdots &\vdots &\ddots &\vdots\\ \frac{1}{p+n}&\frac{1}{p+n+1}&\ldots &\frac{1}{p+2n-1}\end{ ...

Determine all the positive roots of x^{x}=\frac{1}{\sqrt{2}} ...

Find all functions from reals to reals satisfying f(x+y) + f(y+z) + f(z+x) ≥ 3f(x+2y+3z) for all x, y, z Belonging to reals. ...

Determine naturals x and y satisfying \frac{1}{x}+\frac{1}{y}=\frac{1}{14} .....It was done in goiit once! ...

Prove that \tan(\frac{3\pi}{11})+4\sin(\frac{2\pi}{11})=\sqrt{11} ...