Prove that there exists a positive integer $N$ such that there are at least 2005 ordered pairs $(x, y),$ of non-negative integers $x$ and $y,$ satisfying $x^2 + y^2 = N.$

Two friends went to get ice cream sundaes. they each chose a flavor of ice cream from a list of vanilla and chocolate and toppings from a list of hot fudge, strawberries, sprinkles, peanuts, and whipped cream. use the sets below describing their choices and find b'. let a = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream} let b = {vanilla, hot fudge, sprinkles, whipped cream} let c = {chocolate, hot fudge, peanuts, whipped cream} {vanilla, hot fudge, sprinkles, whipped cream} {chocolate, strawberries, peanuts} {chocolate, strawberries} {vanilla, strawberries, peanuts}