2023 AMC 10A Problems/Problem 8

or every real number x, define bxc to be equal to the greatest integer less than or equal to x. (We call this the “floor” of x.) For example, b4.2c = 4, b5.7c = 5, b−3.4c = −4, b0.4c = 0, and b2c = 2. (a) Determine the integer equal to � 1 3 � + � 2 3 � + � 3 3 � + . . . + � 59 3 � + � 60 3 � . (The sum has 60 terms.) (b) Determine a polynomial p(x) so that for every positive integer m > 4, bp(m)c = � 1 3 � + � 2 3 � + � 3 3 � + . . . + � m − 2 3 � + � m − 1 3 � (The sum has m − 1 terms.) A polynomial f(x) is an algebraic expression of the form f(x) = anx n + an−1x n−1 + · · · + a1x + a0 for some integer n ≥ 0 and for some real numbers an, an−1, . . . , a1, a0. (c) For each integer n ≥ 1, define f(n) to be equal to an infinite sum: f(n) = � n 1 2 + 1� + � 2n 2 2 + 1� + � 3n 3 2 + 1� + � 4n 4 2 + 1� + � 5n 5 2 + 1� + · · · (The sum contains the terms � kn k 2 + 1� for all positive integers k, and no other terms.) Suppose f(t + 1) − f(t) = 2 for some odd positive integer t. Prove that t is a prime number.