Asian Pacific Olympiad

Well This problem is from APMO 2009. Just Try This and post your Solutions.

Problem.  Consider the following operation on positive real numbers written on a blackboard:
Choose a number r written on the blackboard, erase that number, and then write a
pair of positive real numbers a and b satisfying the condition 2r2 = ab on the board.
Assume that you start out with just one positive real number r on the blackboard, and
apply this operation k2 ¡ 1 times to end up with k2 positive real numbers, not necessarily
distinct. Show that there exists a number on the board which does not exceed kr.


Well Hope You will like it !!

Well This is the congruency I failed to Solve !

Well This congruency is a good thing to solve. Well it is very easy not a difficult one but still I m posting it !

p^2 is congruent to -11 modulo 36. Well you have to find all integral solution for p and you have to prove that no more solutions exist.

Something Special !

Well What do you mean by a good problem ?
It is the one whose solution should not require any prerequisites except cleverness. A high school student should not be disadvantaged compared to a professional mathematician. Yes, With problems requiring cleverness only with check the mind of the person only not by learning and knowing the whole thing.

Well I came across such a problem last night and here it is for all of you.

Problem. In a Finite sequence of real numbers, every 7-sum is negative, whereas every 11-sum is positive. Find the greatest number of terms in such a sequence. 

 Well The above was an IMO problem of 1977 and it was a problem for 6 points only. It requires nothing but a common sense generating from inside. It is not different for a 10th standard student and not for a professor.

Well Try it And do comment your responses.

Pre RMO

Had My Pre RMO Test Today !!

Well I had It very Good but did some silly mistakes. But I think i will probably clear the cut off of 50/100. There were 8 question in the paper and I attempted 7. The only Question that i didnt attempt is here:

Q- ABCDE is a regular pentagon. A circle is drawn which is tangent to CD at D and AB at A. F is the mid point of arc AD. So you have to prove that AFDE is a rhombus.

Inequalities

This is One Of my Favourites And this is from IMO 1991.

Let I be in the incentre of any triangle ABC then you have to prove that

IA/AD.IB/BE.IC/CF > 1/4.

Get your Proofs Ready.

Number Theory

This is from Number Theory. Might be trivial but its a good one.

Q- If n is not prime and euler function of n is a divisor of n-1 then prove that n has atleast three distinct prime factor ?

Try it.