27th Eötvös Competition Problems 1923

1.  The circles OAB, OBC, OCA have equal radius r. Show that the circle ABC also has radius r. 2.  Let x be a real number and put y = (x + 1)/2. Put an = 1 + x + x2 + ... + xn, and bn = 1 + y + y2 + ... + yn. Show that ∑0n am (n+1)C(m+1) = 2n bn, where aCb is the binomial coefficient a!/( b! (a-b)! ). 3.  Show that an infinite arithmetic progression