Subcontests
(6)x^2_{n+1}= ax_nx_{n+1} + bx^2_n
Let x0,a,b be reals given such that b>0 and x0=0. For every nonnegative integer n a real value xn+1 is chosen that satisfies xn+12=axnxn+1+bxn2.
a) Find how many different values xn can take.
b) Find the sum of all possible values of xn with repetitions as a function of n,x0,a,b. list with powers of 2 - 2019 Ecuador Juniors (OMEC) L2 p5
Bored of waiting for his plane to travel to the International Mathematics Olympiad, Daniel began to write powers of 2 in a list in his notebook as follows:
∙ Starting with the number 1, Daniel writes the next power of 2 at the end of his list and reverses the order of the numbers in the list.
Let us call such a modification of the list, including the first step, a move. The list in each of the first 4 moves it looks like this:
1→2,1→4,1,2→8,2,1,4
Daniel plans to carry out operations until his plane arrives, but he is worried let the list grow too. After 2020 moves, what is the sum of the first 1010 numbers? square inside a square, AE = BF = CG = DH - 2019 Ecuador Juniors (OMEC) L2 p4
Let ABCD be a square. On the segments AB, BC, CD and DA, choose points E,F,G and H, respectively, such that AE=BF=CG=DH. Let P be the intersection point of AF and DE, Q be the intersection point of BG and AF, R the intersection point of CH and BG, and S the point of intersection of DE and CH. Prove that PQRS is a square.