Since the previous board was overcrowded with comments , I decided to create a new note so as to keep this step lively. If this also gets crowded , eventually we will again have new RMO board.What you have to do:
1) Propose a problem in the comments. Then I will add that problem here in this note content by giving the respective credit. And after I add the problem , the comment of proposing the problem must be deleted by the problem poster.
2) The problem poster must not post solution to his/her own problem unless someone from our community posts or no one gets it right even after considerable period of time.
3) Inappropriate / trivial comments are not allowed and must be deleted if posted by chance.An enthusiastic discussion is expected.
4) Please reshare this note so that we can reach most of the Brilliantians.
5)The problems to which solutions are posted will be accompanied by a checkmark at the end.
Q1) Find all prime numbers for which there are integers satisfying and .
Q2) The roots of the equation form a non-constant arithmetic progression and the roots of the equation form a non-constant geometric progression. Given that are real numbers, find all positive integral values and .(Shared by Mycobacterium Tuberculae)
Q3) In an equilateral , is a point inside the triangle such that .Prove that .(Shared by Raven Herd)
Q4) Evaluate with respect to . (Shared by Svatejas)
Q5) Consider a cyclic quadrilateral such that is the diameter of its circumcircle. Construct points and on such that and . Show that .(Shared by Karthik V)
Q6) Prove that polynomial has at least one root in . (Shared by Shivamani P)
Q7) If are positive reals , determine the minimum value of the expression with proof. (Posed by Nihar M)
Q8)a) How many ways are there to represent a natural number as a sum of natural numbers?
Q8)b) How many ways are there to represent a natural number as a sum of non-negative numbers?
Q9) Find all ordered pairs of integers satisfying: .
Q10)a) Prove that if and both factorize into linear factors with integral coefficients, then the positive integers and are respectively the hypotenuse and area of a right angled triangle sides of integer lengths.
Q10)b) Show further that if where are integers, then are numerically the radii of the incircle and the three excircles of the triangle. (Shared by Svatejas S)