# imo

1. ### 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. ### Review my IMO solution

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

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

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. ### 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.