1. H

    Help solving a functional equation

    I am stuck trying to solve the following functional equation: Find all functions f:N_{0}\rightarrow N_{0} satisfying the equation f\left(f(m)^{2} + f(n)^{2}\right)=m^{2} + n^{2} for all non-negative integers m and n. Note: The set N_{0} represents the set of all non-negative integers...
  2. Paze

    Review my IMO solution

    Hi MHF. Can someone please review my IMO solution: 1967 IMO Problems/Problem 1 - AoPSWiki Regards,
  3. K

    Imo schoolarship

    hello, I want to ask you if there is ani organization that gives schoolarships for imo medalists and thanks
  4. FernandoRevilla

    IMO 2011 (Problem 6)

    Let ABC be an acute triangle with circumcircle \Gamma. Let l be a tangent line to \Gamma, and let l_a,l_b and l_c be the lines obtained by reflecting l in the lines BC , CA and AB , respectively. Show that the circumcircle of the triangle determined by the lines l_a,l_b and l_c is tangent to...
  5. FernandoRevilla

    IMO 2011 (Problem 5)

    Let f be a function from the set of integers to the set of positive integers. Suppose that, for any two integers m and n, the difference f(m)-f(n) is divisible by f(m-n) Prove that, for all integers m and n with f(m)\leq f(n) , the number f(n) is divisible by f(m) .
  6. FernandoRevilla

    IMO 2011 (Problem 4)

    Let n>0 be an integer. We are given a balance and n weights of weight 2^0,2^1,\ldots,2^n. We are to place each of the n weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet...
  7. FernandoRevilla

    IMO 2011 (Problem 3)

    Let f:\mathbb{R}\to \mathbb{R} be a real-valued function defined on the set of real numbers that satisfies f (x + y) \leq yf (x) + f (f (x)) for all real numbers x and y. Prove that f(x)=0 for all x\leq 0.
  8. FernandoRevilla

    IMO 2011 (Problem 2)

    Let \mathcal{S} be a finite set of at least two points in the plane. Assume that no three points of \mathcal{S} are collinear. A windmill is a process that starts with a line l going through a single point P \in\mathcal{S}. The line rotates clockwise about the pivot P until the first time that...
  9. FernandoRevilla

    IMO 2011 (Problem 1)

    Given any set A=\{a_1,a_2,a_3,a_4\} of four distinct positive integers, we denote the sum a_1+a_2+a_3+a_4 by s_A. Let n_A denote the number of pairs (i, j) with 1 \leq i < j \leq 4 for which a_i + a_j divides s_A . Find all sets A of four distinct positive integers which achieve the...
  10. S

    IMO Problem II

    This problem comes from this year's International Mathematical Olympiad , comparing with another geometry problem on the next day (problem 4) , this seems a little harder . Enjoy ! Let I be the incentre of triangle ABC and let \Gamma be its circumcircle. Let the line AI intersect...
  11. E

    imo problems

    can anybody please give me a link to a pdf file containing all IMO problems till date! it will be very helpful(Nod)
  12. G

    Really need help on this hard problem IMO

    Let ABC be a triangle such that angle ACB = 135°. Prove that: AB^2=AC^2+BC^2+\sqrt{2}\times AC\times BC I really have no Idea how to solve this, any help is really appreciated. Thanks
  13. Deadstar

    IMO 1960 problem 1

    Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of 11. Anyone know how to solve this one. We write N as abc then... Since N is divisible by 11 we know that a - b + c is divisible by 11. So a-b+c = 0...
  14. M

    Help needed on IMO 1990 question

    (IRN 2) Let S be a set with 1990 elements. P is the set such that its elements are the ordered sequences of 100 elements of S. Knowing that any ordered element pair of S appears at most in one element of P. (If x = (…a…b…), then we call ordered pair (a, b) appeared in x.) Prove that P has at...
  15. P


    Consider five points A, B, C, D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let ℓ be a line passing through A. Suppose that ℓ intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF = EG = EC. Prove that ℓ is the...
  16. J

    Extremely difficult IMO problem

    This problem is from the 2003 IMO, the third level of the American Mathematics Competition. I do not even know how to approach this. "Determine all paris (a,b) such that \frac{a^2}{2ab^2-b^3+1} is a positive integer." The only assertion I can make is that 2ab^2-b^3+1>0.