Frequent Olympiad Stuff Questions

For n >=4 points in a plane , such that distance between any 2 points is an integer, prove that there are at least 1/6 distances , which are divisible by 3 ...

1) Find the last 3 digits of 9999... ...

If you have any question to ask, please ask here. ...

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. ...

*Image* Ans a *Image* Ans d ...

Consider a n*n chessboard. A person moves along the lines of the chessboard such that the sum in the y axis is greater than or equal to the sum in the x axis. Find the number of ways of going from one corner to the other corn ...

*Image* Ans : b ...

http://cheentaganitkendra.blogspot.in/2012/02/indian-national-math-olympiad-2012.html ...

G.C.D of (m,n)=1 1) x ≡3 (mod n) x ≡2 (mod m) Find x. Got it from a frnd. ...

here is a 3 x 3 square.... *Image* how many triangles can be formed with the dots being the three vertices of the triangle?? ...

Determine the largest number in the infinite sequence 1^1/1 , 2^1/2 , 3^1/3 , 4^1/4 , ............ n^1/n ...

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1) Show that [5x]+ [5y] is greater than or equal to [3x+y] + [3y+x] where x,y greater than or equal to 0.. ...

1) Find the last digit of ((5!+9)(6!+4))100! How do we do these types of sums? i think we can't do this by taking modulo 10... ...

P is an interior point of an equilateral \Delta ABC so that *Image* , and BP and CP meet AC and AB at D and E respectively. Suppose that PB : PC = AD : AE. Find \angle BPC . ...

1)Find the number of isosceles triangles with integer sides if no side exceeds 1994. ...

Src: AOPS *Image* ...

Prove that for $$a,b,c>0$$ $$\frac{1}{2a^{2}+bc}+\frac{1}{2b^{2}+ca}+\frac{1}{2c^{2}+ab}\leq \left( \frac{a+b+c}{ab+bc+ca}\right) ^{2}.$$ ...

Here are the problems of RMO 2011 *Image* Source: Mathlinks. ro ...

Find all functions p:\mathbb{Z}\Rightarrow\mathbb{Z} such that p(x^2+1)=p(x)^2+1 . ...

\lfloor\dfrac{r+19}{100}\rfloor+\lfloor\dfrac{r+20}{100}\rfloor+....+\lfloor\dfrac{r+91}{100}\rfloor=546 Find \lfloor r \rfloor ...

You have bought n chocolates from a shop. Now the shopkeeper offers that if you return the wrappers, then for every 3 wrappers, he will give you 1 more chocolate. And you can continue this exchange offer until you run out of ...

A natural number is called "nice" if it is the product of its distinct proper divisors. Find the sum of first 10 nice numbers. ...

ABC is a triangle. The tangent to the circumcircle at A meets the line BC at D. The perpendicular to BC at B meets the perpendicular bisector of AB at E, and the perpendicular to BC at C meets the perpendicular bisector of AC ...

*Image* Find the mistake.... ...

*Image* ...

Not exactly Olympiad stuff, infact I don't know if it is. I just needed a section to post it. Others may try and solve it if they please -: Find the sum of terms of the GP : a+ar+ar2.......∞ where a is the value of x for wh ...

\hspace{-16}$If $\mathbf{\alpha\in \mathbb{R}}\;,$ Then Prove That\\\\\\ $\mathbf{\sqrt{17}\leq\sqrt{\cos^2 \alpha+4\cos \alpha +6}+\sqrt{\cos^2 \alpha-2\cos \alpha +3}\leq \sqrt{2}+\sqrt{11}}$ ...

In a regular pentagon, if the length of each side is a and that of each diagonal is b then find the numerical value of a2/b2 + b2/a2 Note : you are not allowed to use trigonometry in this..!! maybe complex can help ...

The no. of ways of selecting n things out of 3n things of which n are of one kind and alike and n are of 2nd kind and rest are unlike is: ...