Problem 9 · Count two interleaved sequences · 4 points
Made by AI and not reviewed by a real person.
For every positive integer n, Mira makes two dot cards: an n×n square and a staircase with rows of 1, 2, ..., n dots. She sorts all the cards by dot count, keeping tied cards separately; how many dots are on the 100th card?
- 1770
- 1764
- 1711
- 1681
- 1849
Show a hint
- Rather than listing cards from the start, count how many come at or before a conveniently sized square. Remember that equal dot counts still belong to separate cards.
- A staircase with n rows has n(n+1)/2 dots. How many square cards and staircase cards have at most 40×40 dots?
Show the answer and solution
Answer: B — 1764
Up to 1600 dots, there are 40 square cards. The 56th staircase has 56×57/2 = 1596 dots, while the 57th has 1653, so there are 56 staircase cards: 96 cards altogether. The next four dot counts are 1653, 1681, 1711 and 1764. Thus the 100th card has 1764 dots. The tempting choice 1849 results from merging tied cards, but ties must be kept separately.