2017 AMC 12A Problems/Problem 1

Problem

Pablo buys popsicles for his friends. The store sells single popsicles for $$1$ each, 3-popsicle boxes for $$2$ , and 5-popsicle boxes for $$3$ . What is the greatest number of popsicles that Pablo can buy with $$8$ ?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$

Solution

By the greedy algorithm, we can take two 5-popsicle boxes and one 3-popsicle box with $$8$ . To prove that this is optimal, consider an upper bound as follows: at the rate of $$3$ per 5 popsicles, we can get $\frac{40}{3}$ popsicles, which is less than 14. $\boxed{\textbf{D}}$ .

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2017 AMC 12A Problems/Problem 1

Problem

Solution