Subcontests
(4)when binomial is odd then we have a power of 2
Let n be a positive integer. Count the number of numbers k∈{0,1,2,...,n} such that (kn) is odd. Show that this number is a power of two, i.e. of the form 2p for some nonnegative integer p. sum of n different integers divisible by n
(a) For which positive numbers n does there exist a sequence x1,x2,...,xn, which contains each of the numbers 1,2,...,n exactly once and for which x1+x2+...+xk is divisible by k for each k=1,2,....,n?
(b) Does there exist an infinite sequence x1,x2,x3,..., which contains every positive integer exactly once and such that x1+x2+...+xk is divisible by k for every positive integer k? rectangle out of 2 circles & perpendiculars
Let C1 and C2 be two circles intersecting at A and B. Let S and T be the centres of C1 and C2, respectively. Let P be a point on the segment AB such that ∣AP∣=∣BP∣ and P=A,P=B. We draw a line perpendicular to SP through P and denote by C and D the points at which this line intersects C1. We likewise draw a line perpendicular to TP through P and denote by E and F the points at which this line intersects C2. Show that C,D,E, and F are the vertices of a rectangle.